I'm not an expert but I will express my understanding.
The BR will contribute genes for brown eggs. The EE may contribute genes for blue, green, or brown. I think green eggs are a combination of genes for brown and blue eggs. It could be any of these from the EE. The genes for the blue or green egg seems to follow with the genes for the comb. If the chick has a pea comb, then there is a pretty good chance she inherited some genes for the blue or green eggs. Bottom line, I don't think you can predict with any accuaracy until you see the type of comb on the chick, and even then it is uncertain.
I understand there are different ways to make the buff color so there might be something else at play here, but I would expect the offspring of a BR and Buff EE to be the same pattern and color as the father. I think the hens may be a bit darker, but still the same pattern and color. If you cross the offspring of this match, the original BR and Buff EE pair's grandchildren could be any mix of barred or wheaten pattern in white, lemon, gold, buff, maybe more, or a black unicolor. But that is the second generation. The first generation should look a lot like the father in this match.