You must have all four types on eggs in your inventory at the same time. You can get this quickly if you have a friend with all the eggs, just have him give them to you. Otherwise, here are the first instance of their locations. Except the gold egg, they can all be found in multiple spots.
White - Chapter 1-2. Just before you head into the building with the lady in the black dress head up to the roof of the building which has the shotgun. Should be a chicken up there, if not he's already roaming about. Drops as white egg on Normal difficulty.
Brown - Chapter 1-2. Same as "white" but play on Amateur.
Gold - Chapter 3-1. Middle island on the map, the "chicken farm." Arrive and wait a moment, chicken should give a little cock a doodle do and drop it.
Rotten - This drops randomly later in the game from Chapter 4-1 onwards off defeating enemies.