Eggs in Mighty Monsters are obtained through either prize games, or in-app purchases with Diamonds
There are two types of eggs, but all eggs share some similar qualities.
All Eggs are gathered outside of normal questing, all eggs take 2 hours to hatch, and all eggs are hatched using one of your training slots.
Monster Eggs are purchased either individually (Not to my knowledge), or as part of an offer with Diamonds via the in-game Menu (generally with extra Tokens and a piece of Equipment. They will always yield the Monster their egg is named for.
Can yield any Base Epic Monster in the game (possibly excepting the new PVP only monsters).
Can yield any Base Rare Monster in the game.
Can yield any Base Common Monster in the game.