The chicken or egg question is partly just a semantic exercise, but also strays into the territory of evolution and touches on the sorites paradox (also sometimes called "the argument of the beard".)
Semantics: first, phrase the question carefully.
What came first, the egg or the chicken? The egg existed long before the chicken, because many species, including species of bird, produced eggs before there were any chickens.
What came first, the chicken or the egg? Putting it this way round implies that "egg" refers specifically to a chicken's egg. It is therefore more of a "paradox" at first sight, because a chicken must come from an egg that must have been laid by a chicken, which... etc.
However, evolutionary theory tells us that species develop by a series of small changes, the disadvantageous examples of which are selected out by the environment, leaving the advantageous ones to be passed on to the next generation. It is a "numbers game" that relies on probabilities. The most perfectly adapted specimen may nevertheless be killed by a predator before it has reproduced, and the most poorly developed runt may somehow survive to breed. However, across a large population, the species will develop by a series of tiny changes which improve the species' suitability for the environment. This process is then modified and hastened by human agency with selective breeding of livestock.
It is this selective breeding that led to the species that we now call "chickens". Pick any definition of a "chicken" (in terms of genetics, size, weight, colour, behaviour, etc.) and there must necessarily have been a first specimen that completely matched the definition you ahve chosen, although no one knew that at the time.
So there was a first "chicken" according to whatever working definition you choose, and that chicken came from an egg that was laid by a very chicken-like bird, but not by a chicken. That first chicken egg was the result of the mating of a male and female that almost but not quite met the definition of chickens.
This will apply, regardless of how you define the species, chicken. There must have been a first one that met that definition, and its parents must just have fallen short of the definition.
In a sense, that is a semantic argument, but as I once said to a smart-arse customer, "Well, it all depends what you mean by semantics, sir." The argument does not depend on one particular definition of the word "chicken" rather than another, so it is not "just semantics".
Which is where the sorites paradox comes in. The sorites paradox is the paradox of the heap: how many grains of sand make a heap? (Or, the argument of the bard: how many whiskers make a beard?) We can all agree that if we dump a truckload of sand, that will make a heap of sand. We can all agree that a single grain of sand is not a heap.
So, if we look at the truckload of sand and remove one grain of sand, is it still a heap? Of course it is. And if we remove another grain? And another?
Or, if we have a single grain of sand and add another, do we have a heap of sand? Of course not. And if we add a third grain? Or a fourth?
And yet somewhere between the truckload of sand and the single grain, there is a point where we would say, "That is a heap of sand." And yet, would removing one grain from the hep really stop it from being a heap?
Point is, we can all tell the difference between something that is clearly a heap an something that is clearly not a heap, but that does not mean that there is a clear point at which the distinction is made. In the worlds of science, the law, or commerce, this situation is dealt with by arbitrary definitions and limits. That is why we have a 30 mph limit in built up areas, even though a car going at 30.01 mph would not present a greater hazard. That is why we have an age of consent, although few people would argue that a person suddenly and automatically develops a new confidence and understanding of sexual relationships on their 16th birthday.
So, back to the chicken, whatever definition we choose is necessarily arbitrary. Indeed, in cases where birds or animals are bred for show, the governing bodies produce strict definitions of each breed, which are then applied to the specimens that are entered into shows and competitions.
So, is it really the case that there was one specific specimen that was the first one to meet the definition of chicken? Furthermore, was that particular one a blind alley? Did it die before it passed on its genes? What did it mate with — an "almost a chicken"? Was another "first chicken" born somewhere else, just as various pieces of technology were invented independently at different times, in different places? Almost certainly.
EnolaGaia made a valid point: <<Darwin's model doesn't entail any notion there had to have been an "Organism Zero.">> Similarly, there doesn't have to have been a single "chicken zero". Examples may have been born and died without breeding, or may have bred with not-quite-chickens, to create young that were not-quite-chickens. Two full-chickens may have produced a brood of full chickens which all died without breeding.
However, still think that whichever we count as the first "chicken" rather than the first "chicken-like bird" must have come from the first chicken egg.
Incidentally, back to EnolaGaia's point, I heard Prof. Brian Cox commit a grave error on a recent TV show, saying that the fusion of two types of primitive cell was so unlikely that it can only have happened once, and that all higher lifeforms descended from that single common ancestor. Nonsense! There were billions upon billions of primitive organisms, and when there are many billions, even something that is a one in a billion chance is likely to happen more than once.