I believe Lewis meant readers to assume the Scrubb family were adherents of scientism.
There's only a small amount of evidence for this in Dawn Treader itself, but it makes sense in light of his expressed views on scientism in his other work. This interpretation also lends itself to a straightforward reading of Dawn Treader as the journey of an atheist to belief, a journey that Lewis himself made during his life.
WTF is Scientism?
The article "C. S. Lewis: Science and Scientism" by Henry F. Schaefer III quotes Webster's dictionary's definition of scientism as "a thesis that the methods of the natural sciences should be used in all areas of investigation including philosophy, the humanities, and the social sciences: a belief that only such methods can fruitfully be used in the pursuit of knowledge". Schaefer explains further in the following quote:
It is well to note that at least two other terms carry meanings related to the one Lewis intended by the word scientism. The first is “logical positivism,” a system of thought that became popular in the 1920s. The second is “reductionism,” a more recently accepted word that is becoming rather common, particularly among philosophers of science. Taken to the limit, reductionism claims that human behavior is simply a matter of neurons firing in the brain, and the latter can be further reduced to atomic physics. In such a purely reductionist worldview, human responsibility does not exist. Although popular in three different eras, the three terms scientism, logical positivism, and reductionism are sufficiently closely related that distinctions may be subtle. They represent a belief system toward which C. S. Lewis was not receptive. As a more specific example of scientism, consider my relationship with my wife, to whom I have been happily married for more than 30 years. Scientism tells us that if one could make enough accurate scientific measurements on myself and on my wife, the resulting analysis would fully explain my strong attraction to her in preference to all others.
He later cites Carl Sagan, Richard Dawkins, and Daniel Dennett as intellectuals who represent these modern-day versions of scientism.
Lewis was a follower of scientism himself in his younger days, before he had a religious awakening and converted to Christianity. Schaefer includes a quote from Surprised By Joy in Lewis's own words: "You will understand that my (atheism) was inevitably based on what I believed to be the findings of the sciences; and those findings, not being a scientist, I had to take on trust—in fact, on authority."
I gather scientism was fairly popular in English academic circles from the late 19th Century up until about the 1950's, but all my direct knowledge of it comes from reading critiques of its ideas by its opponents, of which it seems to have had many. The picture these works paint of it is not flattering. Followers of scientism in these works are invariably atheists; are usually obsessed with eugenics, genetic fitness, and IQ tests; wish to impose government-sponsored programs to sterilize or kill the "unfit" to preserve the quality of the gene pool; and believe it is their right or duty to subjugate or exterminate "inferior" races. They often support the worst excesses of British imperialism, and after the Second World War, fictional scientism adherents started to take on traits of the Nazis or the Stalinist Soviets.
G.K. Chesterton criticizes ideas that sound rather like scientism in What's Wrong with the World. You can see the profile of scientism in the World State of Brave New World; the Ingsoc party of 1984; Weston, Devine, and the N.I.C.E. from Lewis's own Space Trilogy; and even a tiny bit in Sauron and Saruman of The Lord of the Rings. Objections to scientism in these works are often made on religious grounds. Orwell objects from many of the same ethical beliefs as the others, but his views on religion were complicated. (1, 2)
Evidence from the text of Dawn Treader
Of course, Lewis couldn't write in a children's book that the Scrubbs were atheists who believe in eugenics, imperialism, Marxism, and a Nietzchean rejection of traditional morality. So he settles for a few practices that seem in line with the more harmless things scientism believers might do in their daily lives, but will mark them as odd to most children of the time: they're vegetarians, non-smokers, and non-drinkers.
Even today, vegetarianism, along with atheism and strong faith in science, is associated with left wing politics. Dawn Treader was published in 1952; according to the American Cancer Society, "[t]here were a few small-scale studies conducted from the late 1920s to late 1940s that suggested a possible link between smoking and lung cancer", but strong evidence didn't start to emerge until the 1950's, and the harmfulness of smoking didn't start to become common knowledge until the 1964 Surgeon General's Report on Smoking and Health. So being a non-smoker would have probably seemed very outré to readers in 1952; you'd have to be paying very close attention to the bleeding edge of research to know there was any evidence at all that smoking was harmful. And in my experience, being a non-drinker can mark you as odd in certain social circles even today. The following information about their underclothes, their bedclothes, and their windows being open is probably just meant to further mark the Scrubbs as odd; if I were writing today and trying to achieve the same effect, I might mention that they do all their cooking and make all their own soap with virgin coconut oil, or that they insist every room have a crystal at its northernmost point.
This passage is followed in the book by a short description of Eustace himself:
Eustace Clarence liked animals, especially beetles, if they were dead and pinned on a card. He liked books if they were books of information and had pictures of grain elevators or of fat foreign children doing exercises in model schools.
Voyage of the Dawn Treader, Chapter One: "The Picture in The Bedroom"
He's none too concerned about hurting or killing other living creatures; he likes dead beetles, after all, and he likes them pinned to cards, like an entomologist cataloguing specimens. He doesn't read anything that involves abstraction, whimsy, metaphor, or imagination of any kind; he only likes books of information. He enjoys looking at grain elevators (for some reason) and fat foreign children doing exercises (probably because he's a bully and a weakling, and enjoys watching other people suffer---a bit further on we're told that "deep down inside him he liked bossing and bullying" but that "he was a puny little person who couldn’t have stood up even to Lucy, let alone Edmund, in a fight").
Predictably, Eustace neither believes in nor approves of Narnia:
Last year, when he had been staying with the Pevensies, he had managed to hear them all talking of Narnia and he loved teasing them about it. He thought of course that they were making it all up; and as he was far too stupid to make anything up himself, he did not approve of that.
These passages paint a picture of someone who fancies himself some sort of scientist, carrying around dead beetles on cards as if they were scientific specimens and thumbing through books full of pictures of grain elevators. Eustace doesn't know at this point that Narnia is real, so he thinks it's an imaginative game the Pevensies have made up, and his reaction is cruelty and condescension; not only does he reject imagination and play and belief himself, he also mocks others for accepting them. We're never told straight on that Eustace is an atheist, but this sounds quite a bit like how Christians would perceive someone like Richard Dawkins, who spends a great deal of his time publicly proselytizing atheism and "debunking" religion.
There are little signs in other parts of the book as well that Eustace's family are adherents of scientism. In Chapter 2 we're told Eustace puts down the titular ship the Dawn Treader and keeps on "boasting about liners and motorboats and aeroplanes and submarines"; an obsession with Western society's technological superiority is definitely a trait of the scientism-believing villains in other books, including Weston and Devine in Out of the Silent Planet, who believe the Malacandrian societies are inferior partially because of their limited use of technology.
Later in the same chapter, Eustace writes in his diary that he told Caspian he's a Republican. As an American, I'm assuming he means a British Republican, i.e. someone who supports dissolving the monarchy and forming a republic like the one that existed under Oliver Cromwell, which to me sounds rather like something a Marxist would support, given Marxism's antipathy towards hereditary rulers.
In Chapter Five, Eustace writes about an argument he had with Caspian over whether it was feasible to row back to land on short water rations:
Caspian says the men couldn’t row on half a pint of water a day. I’m pretty sure this is wrong. I tried to explain that perspiration really cools people down, so the men would need less water if they were working.
Ibid., Chapter Five: "The Storm and What Came of It"
I'm pretty sure Eustace is wrong, and that you need to have adequate water during strenuous exercise (such as rowing a large ship); these guidelines for prolonged exercise suggest drinking when you're thirsty and enough to quench your thirst, which the men on the ship wouldn't be able to do. Schaefer quotes Lewis himself from an essay collected in The Discarded Image: "In our age [...] the ease with which a scientific theory assumes the dignity and rigidity of fact varies inversely with the individual’s scientific education". This is exactly what Eustace is doing. He's a child, with no scientific education that we've seen so far, asserting something he thinks is a fact of science when people's lives are at stake, and he's completely self-assured about doing so, never once considering that he might have remembered wrongly or that there might be other complicating factors. His doing this also fits with Faye-Ann Crowell's assertion, which Schaefer also quotes, that "it was not science which Lewis was attacking [in the Space Trilogy] but certain ideas held by people usually not scientists." Lewis has already shown us some of the odious scientifically flavored ideas that Eustace holds, but now he's also showing us that Eustace is not a scientist or even particularly conversant with science despite the airs he puts on.
A Way of Interpreting the Book
The Narnia books treat religion through metaphor and symbolism, so as I said, they never come out and tell us that Eustace is an atheist. But there are enough hints that I think it's a reasonable reading. And if we adopt this reading, then Eustace's character arc in the story is the journey of an atheist to faith, like the one Lewis himself had as a young man.
Eustace starts the story as a non-believer who mocks and bullies others for their belief in Narnia and Aslan. He holds dogmatically to his version of "the real world" even when the evidence of his senses plainly tells him that he's not in that world anymore, that he's in a place where there's no British Consulate he can complain to and no one knows what a Republican is. He treats the people around him with contempt, insulting their intelligence, calling them names, and failing to understand good intentions. He notes in his diary that when he complains about getting too little food or too little water, Lucy is usually quick to share with him, but he doesn't understand why, because charity and kindness are apparently foreign concepts to him: in Chapter Five he writes "Very short rations for dinner and I got less than anyone. Caspian is very clever at helping and thinks I don’t see! Lucy for some reason tried to make up to me by offering me some of hers but that interfering prig Edmund wouldn’t let her." Later in the same chapter, he writes that "Lucy gives me a little of her water ration. She says girls don’t get as thirsty as boys. I had often thought this but it ought to be more generally known at sea." It sounds to me like Lucy tricked him with some made up scientific-sounding flimflam---again demonstrating how little knowledge of science Eustace actually has---to explain her act of kindness in a way that would make sense to him.
Partway through the book, Eustace turns into a dragon, and is surprised when everyone treats him with kindness and sympathy. He starts to understand morality and ethics, and concepts like charity, kindness, generosity, and altruism, as well as developing a sense of camaraderie with the others that was missing before. It's Aslan who finally turns Eustace from a dragon back into a human. We know from the other books that Aslan represents Jesus Christ in Narnia, or rather is Jesus Christ as He chooses to appear in Narnia, so Aslan transforming Eustace back into a human represents the moment that Eustace chose to embrace faith, and through that act was able to end his moment of crisis. The narrator tells us near the end of Chapter Seven that "[i]t would be nice, and fairly true, to say that 'from that time forth Eustace was a different boy.'" After coming to faith, Eustace changed drastically, and became a much more pleasant person, renouncing his previous attitudes and embracing a morality and ethics along with his faith in Aslan / Jesus.
The book ends with Reepicheep, a zealous believer, take a leap of faith to get into Aslan's Country, the Narnian representation of Heaven. Eustace, alongside Edmund and Lucy, come to the edge of Aslan's Country and meet Aslan, who tells Edmund and Lucy that they must find the door to Aslan's Country from their own world. The children return to their own world, where we're told that
everyone soon started saying how Eustace had improved, and how “You’d never know him for the same boy”: everyone except Aunt Alberta, who said he had become very commonplace and tiresome and it must have been the influence of those Pevensie children.
Ibid., Chapter Sixteen: The Very End of the World
After Eustace returns, he's given up the belief system of his parents. He's no longer an atheist; he no longer believes in scientism; he's acquired a sense of morality. Everyone calls it an improvement, except for his mother, Aunt Alberta, who had taught him her own beliefs and isn't happy that he's abandoned them.