From a scientific stance, it would be logical to say that the egg came first, as without the egg the chicken would not be in existence.
Many scientists belive that the chicken egg as we know it, and the chicken itself has changed over a period of thousands of years.
The genetic make-up of the chicken is quite similar to that of the Tyranassaurus Rex, implying that the chicken is a direct descendent. The T-Rex laid much larger eggs, and as it evolved into the chicken as we know it today, the eggs eventually became smaller.
But in all truth, the above statement is a simple theory, but it is quite true that the chicken evolved from a larger animal with larger eggs. This could have been another bird or a giant lizard.
Every animal started as an embryo, which mutates and grows overtime to form an animal of sorts. These animals grew reproductive organs, and eventually gave birth to young in egg form or live young as in the case of mammals.
So the argument cannot be settled simply, as many religious people will present a religious argument, while scientifically minded people will give a scientific argument.
Seeing as it is almost impossible to settle, i'm going to say that the egg came first, as without the egg, the chicken would not come into question. But without the chicken, (or descendent animal) the egg would not be in place...