In all the instances I've been able to think of off the top of my head: how the egg is named depends on whether it's in a culinary/food-centric context.

When you're referring to an egg in its natural state, the egg is named for the animal that produced it, like duck egg, goose egg, etc.

When referring to an egg that has been prepared, or something that is not an actual egg from an animal, you'd refer to it based on the preparation or what the fake egg is made of, like fried egg, boiled egg, faberge egg, or chocolate egg.

I would posit that the question can't truly be answered in its current form; the question is assumed to be asking about a unaltered egg produced by a chicken, but does not explicitly state whether the egg was transformed in some way, and does not state that the egg being referred to is specifically from a chicken.

Yes I'm fun at parties and no I will not be taking questions at this time :)

Edit: Formatting