I got an egg from those chests you can choose from after you're done. Can you get anything good from these eggs or are they like the ones you can find exploring?

I'm not sure if they do differ from regular wild-found eggs, but wild eggs, I'll note, can be any non-fabled (probably those found in Astryl's Vale can potentially be nephs, though) species (probably restricted by what species are or aren't found in whatever area you're in when you find it) of any rarity- which means that if the Gods of Good RNG are in your favour then one could hatch Legendary, or even Mystical.

(eggs found in fest zones, however, will always save glitches be themed pets, and thus at least Rare/2tp)

Eggs rewarded after defeating dungeon bosses tend to have more genetic points than other wild eggs. That's about the only difference.