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[]
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.
Rarity Eggs[]
Rarity Eggs are won playing Monster Slots and can yield any Monster of that Base Rarity, including previous Seasonal Monsters you might have missed out on.
Epic Eggs[]
Can yield any Base Epic Monster in the game (possibly excepting the new PVP only monsters).
Rare Eggs[]
Can yield any Base Rare Monster in the game.
Common Eggs[]
Can yield any Base Common Monster in the game.