I know the squatting is a submission thing, but here's a couple of anecdotes:
I had a golden sex link (now deceased) who became sick. She stopped squatting at the same time she stopped laying due to illness. At first rather than squat she simply ran away. Then she begame more listless, and she did not run away, but she didn't squat either. She never squatted or laid again.
I have a cuckoo Marans who became ill shortly after the sex link died. She stopped squatting that same day, and did not resume squatting again for 6 weeks, although the illness lasted only about 2 weeks. She laid eggs again very shortly again after beginning to squat.
I know it's submission, but in my hens at least it seems to be a specific *kind* of submission, not to be confused with, say, the submission of a lower wolf to the dominant wolf. It seems to be specifically a sexual submission, only when they are sexually receptive.
Interestingly, my hens will do this even for my children - my 6 year old came in from collecting eggs last night, proudly proclaiming that "Sugar did the good-girl squat for me!" so I guess we all count as roosters as far as the chicken is concerned. Either that or she's just terribly confused!