Initially, we can trivially say that all chickens were preceded by dinosaur eggs. This is no fun though, so I'll go ahead and strengthen the "paradox": which came first, the chicken or the chicken egg?
We define a creature as a "chicken" based on its DNA being sufficiently close to the modern species of chicken.
First, we take it as a given that any creature sufficiently chicken-like to be called one will have hatched from an egg. (Indeed, if a chicken-looking creature was born by some other means, we would probably not accept it as a chicken - and this is a proof in and of itself, albeit a less interesting one.) Thus, every chicken is preceded by at least one egg. However, this does not preclude an infinite cycle, which is the source of the paradox.
We next note that, because a chicken does not have the same DNA as either of its parents, and we're defining chicken-ness based on DNA, its possible that a chicken could be born where one or both of its parents are not-quite-chickens. Furthermore, I assert that, since there has been a point in time at which chickens did not exist, this must have happened at least once. If we consider that first chicken, it had the same chicken DNA when it was an egg, so it was in fact preceded by an egg. QED.