I used to think there would be something horrific in them like Maya's stillborn children but the fact that Ayane opened one and didn't scream kinda craps on that. The only other clue about them is that Niki also had a box marked "do not open" and inside were just her unsent letters to Akira. I know that's weak but what else is there?
This actually gives me ideas.
Maya had a box brought to the school before every reset took place up until the 5th reset and the pattern stopped, suggesting that the box or the content isn’t anything figuratively functional. If it’s some relics from the past, it doesn’t have to be brought to school and feels super dangerous to have Sensei carry it since Maya is always sure about the past being his kryponite.
And if it’s anything of sentimental value similar to the unsent letters you alluded to, it might be in the form of farewell letters/last words, like the one we saw when Maya finally succumbed. This might explain why it has to be brought to the school, and not any other places; her way of making sure that these words can live through the apocalypse and reach to him in the event she no longer can’t.