This has been known by many philosophers as a question with no answer,  as long as we refer to a chicken egg not a generic egg (dinosaur eggs existed long before chickens).

If we say the chicken egg existed first then a chicken must have existed to lay it hence the chicken must exist first but how would it have been born except through a chicken egg.

With the concept of evolution however it could be concluded that the chicken egg came first, as the ancestral species of the chicken would have laid an egg which would hatch into a chicken. Even this is open to interpretation as it could be said that the egg is not a chicken egg but rather an egg of the species that laid it or it could be asked how we define a chicken.