Yes, but Egg's aren't meant to level up or have items affect them - they don't in the main games, and it doesn't make sense either!

Also, discovered that you can use items on eggs as well - the Iron stat booster boosted the egg's defence...
EDIT:
Forgot to mention they also seem to hatch with a copy of whatever the mother/father was holding (I only ditto breed so presuming non-ditto 'mon). For example, a Totadile holding a Oran berry bred with a Ditto gives an egg which hatches into a Totadile holding an Oran berry. This shouldn't happen. I could easily see people breeding quick-hatching 'mon and expoiting this for infinite amounts of rare items such as Rare/Gold Candy, Nuggets, etc.