Q: How do brown eggs become brown (instead of white)?
A: The brown color in an egg is laid on in the chicken's reproductive tract by the shell gland pouch. Some breeds lay a tinted egg with a very light brown color, while others lay extremely dark chocolate brown eggs.
From right to left: Marans egg, Welsummer egg, Rhode Island Red egg.
Within a breed, too, hens can lay eggs darker or lighter than the average, depending on whether they produce more or less of the brown color that goes on the shell--and how long the egg stays in the "paint station" of the reproductive tract.
Interestingly, individual hens--while they lay basically the same color throughout their lives--will still start out laying darker eggs at the beginning of the season after their bodies have had a (winter) break. When they hit their egg-laying stride, laying as quickly as they can for their breed, often their eggs lighten up. There is only a certain amount of color produced. If it is laid over more eggs or over a larger surface area--or if it moves more quickly through the shell gland pouch--the eggs will be a lighter color. (Heat stress and some illnesses can also cause eggs to lighten.)
Even the egg color of dark brown layers like Marans or Welsummers will naturally vary in intensity over the course of the season. The eggs of Marans, for instance, can go from deep reddish chocolate to not much darker than the egg of a regular brown layer. After a break in laying such as a molt (or a period of broodiness), the eggs will become darker again.