Cooking an egg is endothermic because when you cook an egg it absorbs the heat from it's surroundings like boiling water or heat from a pan to cook the egg.
This means that the heat energy is taken in to denature the proteins that are in the egg and cause teh liquid to solidify into a cooked state.
Essentially the egg is taking in the heat to change it's structure.
The primary chemical change when cooking an egg is the denaturation of the eggs proteins.
This involves breaking the bonds that hold the eggs protein molecules in their original shape and the process requires heat energy to be absorbed from the environment.
As the egg cooks, the egg then draws heat from the pan or the water that it is in and causes the temperature of the cooking medium to decrease slightly.
And unlike an exothermic reaction, the cooking process of an egg does not release any significant heat back into the environment.
Endothermic chemical reactions are those in which the reactants take in heat energy from their environment to create products.
The process of boiling water is also endothermic because heat must be applied in order for the water to boil.