What gives them the green eggs is definately a brown gene in there, you are correct.
Many hatcheries bred other breeds into their ameraucana flocks to increase production, and possibly add in new genetics, that is what causes some color differences and egg colors. I ordered up about 50 Ameraucana's last year from different hatcheries, 90% of them laid brown eggs. To me that was very dissapointing. So that is one reason I am so against people breeding in other breeds and calling them easter eggers, to me if you get an easter egger it should lay blue to green to olive eggs, NOT brown.
I havent found much difference in the health between the two, I am getting new lines of Ameraucana's every year so that helps to keep my gene pool a but more broad.