Eggs certainly came before chickens, but chicken eggs did not—you can't have one without the other. However, if we absolutely had to pick a side, based on the evolutionary evidence, we're on Team Egg.
What came first the chicken or the egg psychology?
What came first - the chicken or the egg? Chickens are hatched from eggs so the egg must have appeared first. But eggs come from chickens so the chicken must have appeared first.
Which existed first chicken or egg?
So in a nutshell (or an eggshell, if you like), two birds that weren't really chickens created a chicken egg, and hence, we have an answer: The egg came first, and then it hatched a chicken.