Science does in fact have an answer to the oft-repeated conundrum, Which came first, the chicken or the egg? This of course means that you should never let anyone offer this as a riddle ever again without correcting them.
In nature, organisms evolve through small changes in their DNA. This is because DNA copying during reproduction is never 100% accurate, and these minor mistakes in copying are the fuel for natural selection. Beneficial mutations that allow an organism to better pass on its DNA are naturally selected by various environmental pressures (sexual, predatory, etc.), and can eventually become dominant in a population of organisms, resulting in new species.
In an animal like a chicken, DNA from a male sperm cell and a female ovum (egg) meet and combine to form a zygote: the first cell of a new baby chicken. This first cell divides innumerable times to form all of the cells of the complete animal. In any animal, every cell contains exactly the same DNA, and that DNA comes from the zygote.
Chickens evolved from non-chickens through small changes caused by the mixing of male and female DNA or by mutations to the DNA. These changes and mutations only have an effect at the point where a new zygote is created. That is, two non-chickens mated and the DNA in their new zygote contained the necessary mutation(s) to the embryonic body plan to make the first true chicken. That one zygote cell then divided and formed a biologically modern chicken.