I am tired of hearing this dumb question on Dennys commercials, etc. I thought this out long ago; 1970's?
1. As we were taught in High-School: first define the problem: What is a chicken? Well, a chicken is about yo big, has two legs, two wings, etc. We provide a complete physical description of a chicken, both internal and external, down to the cellular level. Note color, size, and other variations between different kinds of chickens, etc.
2. Where does this set(s) of physical characteristics
come from? From the DNA (genes).
The physical characteristics MAP to the DNA; so we now have a 'SET' (as in a mathematical set) of genes which define a chicken or all kinds of chickens.
3. Let's say we have a "beast" which is almost, but not quite a chicken; it has a beak like a duck, and webbed feet. We can take a scissors and cut the webbing from its feet, and grind its beak down until it looks just like a chicken. But it is NOT a chicken because it still has its "beast" DNA. Can we alter the DNA to make it a chicken? No, because we would have to alter the DNA in every single cell of the beasts body. But wait! There is a point in every animals life when it exists as a single cell: immediately after fertilization of the single-cell egg.
At the single-cell stage the DNA can be altered by chemicals, radiation (man made or natural), viruses, other mechanisms, or "natural mutation" which means that we don't know what caused the DNA to change. In the case of genetic damage there exists in each cell a helper molecule that attempts to detect and repair damaged DNA (I read it in a newspaper or magazine), but may not always repair the DNA sequences exactly.
So at the time when the beast is a single fertilized cell one of the above events changes the beast DNA into genes that fall into our SET of chicken genes. The single cell reproduces accurately and ALL the cells have the same DNA. The beast lays an egg which hatches into a chicken.
The EGG CAME FIRST.
Or, the short, non-technical, but logical as Spock answer:
In order for the chicken to come before the egg, the chicken would have had to been born LIVE, not from an egg.