I completed the dungeon for around 10 times but i never even got a Mission Egg… 
When you kill a boss on the 28th floor, you get an egg. Work at least I got.
I always Killed the Boss
Sry a mistake^^
The First time i got one but then never after 
because you can win the price only once my friend.
Oh
Go to the infinite dongeon there is gold egg every 10 floors (9-19-29…)