Any clue if the offspring of a barred rock rooster and EE would produce blue/green colored eggs?
There is one gene pair that determines the base color of the egg. Since it is not a sex linked gene both the father and the mother have two genes at this gene pair. The blue egg shell gene is a dominant gene, which means in the absence of any blue gene the base color will default to white. If just one of the genes at that gene pair is blue, the base color will be blue. That’s about as basic and simple as anything in chicken genetics gets. The base is either blue or white.
Now it gets messy. Brown and green are simply brown coloring on the base color. There are a lot of different genes that affect shades of brown. Some are dominant, some recessive, and some only act if another specific gene is also present. That’s why you can get so many different shades of brown and green eggs, there are a tremendous number of possible combinations. They can interact in really strange ways. There is even one gene that if it is present it bleaches out some brown gene and can turn the egg white or blue.
So to simplify it:
Base blue + no brown = blue
Base blue + brown = green
Base white + no brown = white
Base white + brown = green
Your Barred Rock rooster is not going to have any blue egg genes so his offspring will not get any blue genes from him.
If your EE is laying a white or brown egg, her offspring will not get any blue genes from her. She doesn’t have any to give or her eggs would be blue or green.
If she is laying blue or green eggs, she either has one blue egg gene at that gene pair or she has two. If she has two she will give one to all her offspring so all her daughters will lay either blue or green eggs regardless of what rooster she is crossed with.
If she only has one blue egg gene, some of her offspring will get a blue gene, some will not. So if she is crossed with your Barred Rock rooster, some of her daughters will lay brow eggs and some will lay blue or green.
So if your EE is laying a colored egg, there is a good chance at least some of her daughters will too. If she is laying a brown or white egg, none of her daughters will lay a blue or green egg with that BR rooster.