Borchert wrote Das Brot (The Bread) in 1946 and Die Küchenuhr (The Clock) in early 1947. Both stories involve a man eating bread at half past two. In English, the translation "half past two in the morning" is a bit misleading, since the German phrase "halb drei" says neither "Morgen/morgens" (morning/in the morning) nor "Nacht/nachts" (night/at night). However, the word "Nacht" (night) or "nachts" is repeated several times, not just in these two stories, but also in other ones, such as Nachts schlafen die Ratten doch" (Rats sleep at night) and "Das ist unser Manifest" (This is our manifesto).
I am not aware of half past two having a special meaning in the context of World War II or the first post-war years. Wikipedia and several other online sources say it refers to the darkest and coldest part of the night. Note that the men in both stories try to deceive someone about what happened at that time during the night. In "The Bread", the husband claims that he came to the kitchen because he thought he had heard something, while in reality he had surreptitiously cut off a slice of bread. In "The Clock", the anonymous man lost his parents and everything he had in a bombing. He seems to try to deceive himself about the fact that the bombing took place at half past two, causing the clock to stop, because that was also the time when he always came back home and ate the bread that his mother prepared for him. Strictly speaking, we don't know when the bombing took place, but the man tells the people sitting on the bench that they shouldn't talk about the bombs and the "clock" ("Küchenuhr" literally translates to "kitchen clock"), the only thing he could save, is described as a "Telleruhr" (literally "plate clock") because it is white and has the same glazing as the plate from which he used to eat his bread.
So "half past two in the morning" is a figuratively dark hour in both stories. In "The Bread", the contrast between darkness and light is also used at the end of the story, when, the next evening, the wife tells her husband he should eat four slices of bread instead of the usual three: it is only a little while later that she sits down at the table under the lamplight.