You have the correct answer, echoed by others. If one could draw a sharp line, then we would see proto-chickens laying eggs containing chickens and then some of those chickens laying eggs containing proto-chickens, back and forth for many generations. The combination of alleles that qualifies an organism as a chicken would arise and then often be reversed by recombination in the next generation. Eventually as more and more of the population has chicken allele combinations, the percentage of chickens born would grow as the reversals became less numerous than the forward conversions.

Now, what defines whether an egg is a proto-chicken egg or a chicken egg? An egg is formed by the action of maternal genes, so it will have the characteristics given to it by the proto-chicken mother. But if you just define an egg that hatches a chicken as a chicken egg (rather than an egg laid by a chicken), the egg always comes first.