By his own, Milton means the apostles. The context of both Matthew and Mark makes this clear. The Pharisees ask Jesus about divorce. Jesus replies that given what Moses said, divorce cannot be regarded as entirely forbidden. Adultery in particular is grounds for divorce. However, Jesus says to the Pharisees:
Moses, because of the hardnesse of your hearts, suffered you to put away your wiues: but from the beginning it was not so. (Matthew 19:8)
The godly attitude would be to soften one's heart—to relent even in cases of adultery, and to find a way to reconcile. Jesus goes on to say that remarriage after divorce constitutes adultery:
And I say vnto you, Whosoeuer shall put away his wife, except it be for fornication, and shall marry another, committeth adultery: and whoso marrieth her which is put away, doth commit adultery. (Matthew 19:9)
The apostles thereupon say to Jesus that perhaps it is better to remain celibate if marriage could lead to such difficulties: putting up with an adulterous spouse or, by divorcing and then remarrying, committing what is in effect adultery. The assumption here is that marriage, divorce, and remarriage are all caused by sexual urges, and it is better to forgo the satisfaction of those urges altogether than to risk either countenancing or committing the sin of adultery. Jesus responds:
All men cannot receiue this saying, saue they to whom it is giuen. For there are some Eunuches, which were so borne from their mothers wombe: and there are some Eunuches, which were made Eunuches of men: and there be Eunuches, which haue made themselues Eunuches for the kingdome of heauens sake. He that is able to receiue it, let him receiue it. (Matthew 11–12)
That is, Jesus agrees with the disciples that celibacy is preferable to marriage. He says, however, that only eunuchs can lead a celibate life. Eunuchs are of three kinds: those born so; those made so by others; and those who make themselves eunuchs (i.e., swear celibacy) for the sake of the kingdom of heaven.
Mark has a less full version of these exchanges between the Pharisees, Jesus, and the apostles, but does make clear that Jesus is speaking to the apostles when he forbids divorce:
And in the house his disciples asked him again of the same matter. And he saith unto them, Whosoever shall put away his wife, and marry another, committeth adultery against her. And if a woman shall put away her husband, and be married to another, she committeth adultery. (Mark 10:10–12)
Milton has these passages in mind in the sentence you're asking about. Milton notes that Jesus endorses Moses's words, i.e., that a man may in fact divorce his wife. Jesus however says that the Law allows divorce (and remarriage) only because humans are too weak for the alternatives: either forgiveness and reconciliation, or giving up marriage altogether and remaining celibate. To his own disciples, however, Jesus urges those alternatives. That is, Christ deny’d divorce to his own: he told his apostles that divorce breaks a sacrament and is permitted only as a concession to human fallibility. Milton goes on to say that unlike the apostles, we are "unregenerate" and (being more like Amy Klobuchar than like Pete Buttigieg) have "not attain'd such perfection". For us lesser mortals, then, divorce is of necessity permissible.
Edit in response to comment. @DLosc asks:
Is Milton saying that Christ forbade divorce, but only for some people (those who have attained perfection)? Or is he saying that Christ in theory forbade divorce, but none of us are perfect, so we can't live up to that standard?
Milton interprets these passages to mean that though Christ did not approve of divorce, he nevertheless permitted it. Christ says to the apostles that in an ideal world, there should be no divorce. Since the apostles are Jesus's own, they must abide by the ideals and honor the proscription against divorce. But, Milton asks,
what is that to the unregenerate, who hath not attain’d such perfection? Let not the remedy be despis’d which was giv’n to weaknes.
Unlike the apostles, the mass of humankind is unregenerate. Since we have human weaknesses (such as hard hearts), the Mosaic law allows for divorce. Jesus does not overturn Moses, which forbidding divorce outright would do. Instead, he says that Moses allowed divorce only because he acknowledged our imperfection. Milton says that divorce may be a remedy for weakness, but it should not be despised on that account.
In my holiness-movement, born-again Christian background, "his own" and "regenerate" would apply to all Christians, though "attain'd such perfection" would only apply to some.
In the context of this passage, his own refers only to the apostles. Milton is not arguing that Christ forbids divorce to all Christians, and that it should therefore be available only to unregenerate non-Christians (such as, presumably, the Pharisees). Rather, Milton is making the case for reconciling divorce with Christian belief. So what were Milton's beliefs?
Milton is generally held to be an Arminian, which would place him in opposition to the Calvinist theology of the Puritans. Stephen Fallon describes the difference between Arminianism and Calvinism as follows:
The debate between Calvinists and Arminians rests on the question of free will: for Calvinists, God chooses whether human beings are saved or damned before creation and irrespective of their choices; for Arminians, God predestines human beings only as he foresees their free choices in response to sufficient grace. Predestination is normally understood in its "absolute," Calvinist form, as referring to God's division of human beings into the saved and the damned prior and without regard to any demonstration of merit. In Arminius's "conditional" predestination, on the other hand, God does not select particular human beings (as opposed to others) for salvation; instead, God decrees that those who freely accept universally offered grace will be saved.
A key word here is sufficient. In Arminian belief, God's freely offered grace is both necessary and sufficient for salvation. This contrasts with the Calvinist position that God's grace irresistibly draws only the elect; i.e., grace, while necessary, is not sufficient, as the elect are predestined to salvation.
For Milton, then, divorce might be a sign of weakness, but it is not a sign that divorced individuals have turned away from God's grace, which is freely available to all. The unregenerate here does not mean damned, or un-Christian; it simply means those who do not share the moral stature of the apostles.