What is egg color?
Lets start this off with what is egg color, or how it gets there. The brown egg color is actually applied on top of the shell...if you don't believe me, just run your fingernail over a brown egg, I bet you can scrape some of the color off. This coloring is applied in the abdomen of the chicken before the egg is layed. A combination of genetics determines how dark or light the coloring will be. A white egg is absent of this brown coloring added, it is simply the egg shell. A blue egg, is like the white egg. It is the color of the shell itself with the absence of any brown on the outside of the shell. A green egg on the other hand is a blue shell with the brown color applied to the outside. This happens when a blue egg laying chicken (ie Araucana) is crossed with a brown egg laying chicken, commonly seen in the hatchery mutts called Ameraucanas or Easter Eggers.
Why breed for color?
Well, it really does not matter what color the eggs are, as far as quality of nutrients in the egg, or shell quality. It really comes down to personal preference. Many of the people that buy eggs from us prefer to see the brown and blue/green eggs because it creates a color difference from the mass produced white eggs found in the grocery stores. Some people really like blue/green eggs, because they are pretty and there are some myths about them being lower in cholesteral. Like I said it is all personal preference.
Why cross the breeds in the first place? How does this concept of hybrid breeds work?
Many chickens are inbred. If you are familiar with inbred animals, you will notice that they are weak and less productive. This same concept is true for chickens, when you cross the breeds, it introduces new genetics and if they line up just right (called nicking in the horse world), you will end up with an exceptional animal. You however should not use the offspring of the hybrid chickens for breeding, the reason being you will end up with bad traits showing up, the first generation is always the healthiest. It is entirely different story when you are developing a new breed though. All in all, if your breeding is successful, you can make more efficient egg layers.
How to get the color you want?
You always need at least the male or the female to lay the desired egg color when breeding for blue, green or brown eggs, where when breeding for white eggs, you need have both the male and the female to lay white eggs.
Green or Blue EggsThe blue color is dominant, so breeding a blue egg layer to a breed that lays another egg color will get you another blue or green egg layer (unless blue egg laying bird was heterozygous for that trait then there is a chance of getting a bird who lays a different color egg such as brown or white). So in other words, your odds a pretty good of getting a blue egg laying chicken. If the chicken layed actual blue eggs and you breed her to one that lays brown eggs then you will get a green egg layer, however when the blue egg layer is bred to a white egg layer, you will retain the blue color without adding any brown on top which creates the green color. Sorry if that gets a little muddy in there...sometimes putting forth all those hypothetical scenarios gets confusing. If you are confused, just ask a question in the comments and I will try to clarify.
It is also thought that the green/blue egg color is linked to the peacomb of the Ameraucana and Araucana breeds, however it is possible for these two genes to separate as I have had a blue egg layer with a single comb.
Brown EggsThis is a dominant trait, so as long as one gets passed on to the offspring, they too will lay brown eggs. This allows you to breed a brown egg layer to a white egg layer and get a brown egg layer. Some of the "red sex links" are bred this way, with the white chicken being a white leghorn. This is really pretty simple, what gets more confusing is what shade of brown they will lay. It is passed down from the parents, but it is much more complex. Breeds that have been laying dark brown eggs like Marans and Welsumers will keep doing so, but when bred to a light brown egg layer, they could lay something in between, dark eggs or light eggs.
It may not seem this way, but white eggs are actually a recessive trait, so they must have acquired it from both parents. This is why when people are breeding show Leghorns and such, they will throw out any bird that lays tinted eggs, because it will cause their pure white egg layers to lay tinted eggs as well.
So there is my quick little post on breeding for egg color.