I'll be honest.. I haven't read through past the first page of this thread, so if I repeat anything? My apologies!
But, yes, whatever you feed your chickens can affect the taste.
That being said, I don't think flax seed will affect the taste too much, but you should vary it a bit. I have some girls who won't touch flax seed with a 10ft pole and other's go nuts on it! Fruit (none citrus) and veggie scraps will be great boosters. And Black Oil Sunflower Seeds (aka BOSS) to help keep the weight on during the winter months.
Anything garlic or onion? Unless you really, really like that taste? Don't feed it to them! LOL Even a smidgen of that comes out a bit strong in my girls end product.