There are two genes at each gene pair on the DNA. When the egg is fertilized it gets one gene from each parent. If both genes from one parent for egg color are blue, then a blue gene will be passed down. If both genes from the other parent are white, then a white gene will get passed down. If the parent has one blue and one white, then either gene might get passed down, totally randomly.
The brown egg shell color is a lot more complicated. Instead of just one gene pair affecting brown there are many different gene pairs involved. Some of these genes are dominant, some recessive, some only act if another specific gene is there. That’s why you can get so many different shades of brown, there are a tremendous number of combinations of these genes. Pullets from the same parents can lay wildly different shades of brown or green eggs, just depending on which of these “brown” genes go passed down and how they combine.
I think you are talking about the genetics of your cockerels. If one parent was a true Ameraucana then the cockerels have one blue gene from them. If the other parent is a Marans, they received a white gene from that parent. It does not matter which parent was mother or father, that cockerel will have one blue and one white. When he fertilizes an egg, he will randomly pass down one of those. You don’t know which one.
Brown again is a lot more complicated. Blue is an absence of brown because if brown is present on blue you have green. To simplify it, assume your original Ameraucana had no brown genetics at all. That’s not always totally true because one of the genes that affects brown is a “bleach” gene, it can cause one brown gene to have no effect even if it is present. From some of the egg shell colors I get I think I have that bleach gene in my flock. This stuff can make your head swim, at least it does mine. Anyway, assume that cockerel inherited no brown genes from its Ameraucana parent and inherited brown genes from the Marans parent. Which brown genes. I don’t know. You can reasonably expect that it has a lot of different grown genes from the Marans at various gene pairs and not-brown genes at those gene pairs from the Ameraucana.
Which of these genes will it pass down at each gene pair? It’s totally random. It could be a lot of brown, it could be a lot of not-brown. Remember there are a lot of possible brown genes so there is a tremendous chance it will pass down some brown, but the results could be all over the place as far as shade. Of course you have to consider whet the hen she is mating with pass down.
If you are really interested in studying egg shell genetics you can read this thread. Follow the link in Tadkerson’s post #15.