A case can be made for egg, because the immediate ancestors of chickens laid eggs. One could even imagine the very first chicken hatching from an egg laid by something that was almost, but not quite, a chicken. The problem is that the boundary between not-quite-chickens and barely-chickens is not so sharply defined. And the idea of a first chicken is problematic because whatever offspring it had would have been the result of it mating with with a not-quite-chicken, meaning some of the first chicken's offspring could be chickens, and others not chickens. It could even happen that ALL of the first chicken's offspring are not-quite chickens, so that there's a gap between the first and second chicken. It's also possible that the first bird that meets the definition of "chicken" might not produce ANY offspring, such that all chickens trace their lineage back to a later, independently mutated second (or third, fourth, etc.) chicken instead of the first. And then there's a problem with definitions. Do we define "chicken egg" as an egg layed by a chicken, or as an egg from which a chicken hatched?
I think it was someone on this forum who made an analogy that this is a bit like trying to identify the first person to speak French. It would take a pretty contrived definition of "French" to be able to distinguish the very first french speaker from all of the almost-French speakers with whom he communicated just fine.
So I suppose the best answer is that there is neither a first chicken nor a first chicken egg, but just a very fuzzy transition involving many individuals in a very un-smooth progression in which individual offspring could be either more, or less, chicken-like than their parents, but with chicken-like features becoming more frequent in the population.
On the other hand, the phrasing of the question just said "egg", and not "chicken egg". That loophole makes "egg" the correct answer, even if it isn't the most explanatory answer, or the answer that you're looking for.