Question about eggs in bread

I read last week that 1 tablespoon of Chia seeds to 3 tablespoons of water can be used as an egg substitute.

I haven't tried this myself but presumably you have to soak the seeds and then put the mix in a blender, or mash it into a paste. Maybe someone with a bit of knowledge about vegan foods will have more information on this.

Otherwise, I'd say a couple of days before it turns furry.

If anything, the extra fat from the yolk will delay staling a bit.  The milk and eggs give you a softer, fluffier bread. 

Enriched bread will stay fine for 4-5 days, depending on your room temperature and humidity.  I keep mine wrapped in a clean towel or plastic, and stored in a bread box.  Pain de mie will last up to a week (will be stale, obviously, but not spoiled).  I will never know what the shelf life for babka and sweet buns is, because they disappear too fast :).  I understand you want to be careful with storing dairy at room temperature, but if it's been incorporated into the dough (not baked separately as a custard, for example) it is perfectly safe.  If you want to extend the shelf life of your breads even more, consider using sourdough instead of commercial yeast.