Logically speaking, the egg itself has to come from the female of a species, then fertilized by the male of said species, so egg itself would not be considered a chicken egg unless the parents were chickens. Otherwise animals would have an entirely different kind of offspring inside them. Even if there was a successful next generation mutation between species (which there hasn't been any since mutations are always harmful to the host in some way) or even between several generations, if a dinosaur, a predator, somehow birthed a pseudo-chicken, a prey, it would not end well. Why would anything stronger than a chicken not last longer than a chicken? The chicken would most surely be eaten, or some other natural selection event would take place. The fact that chickens are alive today is a miracle in itself, considering how their intellects and probability of survival are.
As for the "lion+tiger=liger" example, that can not be considered a natural phenomenon, but an interaction produced by the intervention of man, something with intellect that breeds two animals with similar genes to produce offspring with maybe one or two beneficial mutations, but is generally genetically unstable. Same with the mule. A horse and a donkey produce a stronger and better animal, but it's born sterile, and cannot continue to the next generation on its own with out intervention by man. These two pairs of animals do not otherwise produce offspring on their own in the wild, so why would it happen in the past where man supposedly did not even exist yet?
Unless you want to pull in that aliens were the force that created chickens, which is as far fetched as creationism apparently, there is no way two animals that do not have a natural mating pattern together, end up having offspring that are successfully mutated and are able to last longer than their much stronger predecessors.
So long story short, the chicken came before the egg.
