The Chicken And Egg Problem, this problem refers to all species

If the problem is considered specifically w/r/t 'chickens', then
eggs were a form of reproduction in many lineages long before anything even looked like a chicken.The first 'chicken' would have had to hatch from some 'proto-chicken' egg, and therefore the egg would have come first.If the problem is considered more figuratively, with the chicken representing higher organisms in some general way, then the 'chicken' would have come first, simply because egg laying is a highly-evolved form of sexual reproduction (oogamy) that evolved through various intermediate stages of anisogamy (fusion of gametes of unequal size).
The ancestral state of sexuality in multicellular organisms is isogamy (gametes of equal size fuse) - there is no overt sexual differentiation in such species - only '+' and '-' mating types. Here is a good reference on the evolution of sexuality written by one of my former advisors some years ago.

If you don't like my reasoning, you can try Zhang's.Click here and scroll down.

Since RazD has raised a question of some actual consequence,
I am lured back into this thread.An 'egg' in the strict biological sense, is a single germ cell with the capacity to give rise to an embryo.
However, with the exception of mammals and a few fish where development is nurtured internally by the female, the egg also contains all the resources necessary for embryological development of an entire new multicellular organism.
The biological condition of egg production is termed 'oogamy'.It doesn't matter whether it is sexually or asexually produced, but it must contain sufficient resources for completed embrylogical development if it is laid freely in the environment, so many other forms of dispersed propagule (say a fungal spore) do not qualify because they require external inputs of energy for developmenting into a multicelluar organism.However, I would contend that the evolution of the egg was contingent on the evolution of sexual reproduction first, even though some lineages secondarily lost sexuality and evolved eggs that could develop successfully without fertilization.The evolution of sexuality probably began with the fusion of gametes of equal size (isogamy), but the emergence of the 'male' stragey of producing the smallest possible gametes resulted in the countering 'female' strategy of investing sufficient resources in each to ensure the survival of the zygote, regardless of the sperm containing virtually zero resources.As I stated in another thread: