The Egg has a 1/4 chance to be dropped from a chicken. The Farm Chicken from the Barn also produce them, in way higher quantities.
Eggs are used for recipes of 1 in Cake and Carrot Cake, 40 in Omelet, and 5 Pancakes.
Fire Eggs and Ice Eggs come from Fire Chicken and Ice Chicken respectively. These can only be used to in 40s to make Fire Omelet and Ice Omelet.