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not so much, lots of EEs can lay more blue than a purebred APA Ameraucana or Araucana. My experience anyway.
crossing with different breed, blue gene is dominant but then thee gg will be painted with braun too. unless white, then looks pale blue but can be bred back with easily.
A well-bred Ameraucana/Araucana would never lay a green egg. That's a matter of the quality of the stock.
Chickens have two 'slots' for egg color. And those can be white or blue - blue being dominant. The way I understand it, brown has it's own seperate thing and is actually an "overlay" of color.
EE's are a mixed pot and unless you know their background, so if you get EE's it is safest to assume they are likely heterozygous for blue eggs (one blue, one white) or could be homozygous (blue, blue), but have brown overlay (green) if they lay blue/green eggs. I'm not sure how hard it is to get rid of brown overlay in birds, never tried. And some EE's will still display Ameraucana characteristics (muffs/beard/green or slate legs) but lay brown/pink eggs (white/white with overlay).
The problem is that when you cross with another breed, if it is not Araucana, Ameraucana, or EE then the bird is pure for white/brown eggs (white,white). So, say your hens are heterozygous (blue,white). The resulting chicks would be -
(blue,white) (blue/white) (white/white) (white white) Using a punnet square. Therefore, 50% of your chicks would lay brown/pink/white eggs.
If your hens were homozygous, which seems unlikely unless you know their lineage, and you crossed with an outside breed - then all the babies would lay blue eggs, but all would be heterozygous.
True, I do have an EE hen of unknown lineage that lays very blue eggs. But the others have layed green/olive.