Blue is dominant over white. With a green egg laying bird, then, you may or may not pass along the blue egg gene to the offspring, because your EE may have two or one copy of the "blue egg" OO gene.
As for brown, I guess the genes for that are quite complex, on many different loci, and so far as I know, they haven't entirely been figured out, yet. Some are thought to be sex linked, to make it even more difficult to study, and there are intensifying and diluting genes, as well. If brown is passed along, it may be a slightly different color brown. You never know, though. There are many "brown" genes, but maybe none of them would be passed along and you would get white.
Since green is a mixture of both blue and brown (in an eggshell at least
), my GUESS is that you'd either get green/turquoise or some tint of brown. I don't think it's likely to get plain blue, but it's difficult to tell what genes your current bird has. I will say that the pea comb is closely tied to blue-egg laying, so if you get a pea-combed pullet from a green egger, she's likely to either lay blue or green, not "pinkish" or brown.
(I don't know if that helps, or is even more confusing...)
