Out of my four hens, I have two that always squat, one that does sometimes, and one I can't generally get near enough to see what it will do, lol. The two who squat are not the most submissive in my flock, but they are the least skittish. In fact, one of them is the alpha of the flock. They are also the two that lay eggs every day. The other two are not so reliable.
Though I suppose that the squatting might be a sign of submissiveness to an owner or a rooster, I believe it probably has more to do with their hormones, thus their brooding and possibly even their mothering capabilities. I have never seen hen squat in submission to another hen, that is for sure!
My mother breeds dogs, and the ones that most eager to present have also been the ones that made the best moms, both from a fertility standpoint (always producing lots of healthy puppies) and as pertained to their willingness to nurture their babies for longer periods of time, nursing, teaching, etc. The dogs that don't present as often, or who dislike it altogether, have made worse moms, or sometimes not conceived at all (even after multiple Artificial Inseminations...)
I believe chickens have real submissive signs, such as head ducking, pecking order at the feed, even roosting order. Just like dogs: I have never seen a female dog present to another female (or male when she's not in heat for that matter) as a sign of submission. But tail, eye and ear position,who eats first, yawning and bowing: these are all signs of submission. I think that because we humans cower in submission, we assume that a similar position in an animal must be the same thing, but it's just not always true. That said, I am no chicken behavior expert, and since I don't have a roo anymore, I can't really test the fertility/mothering link myself. Someday!