Squatting is a signal that a hen is ready to lay eggs and usually, the eggs will come soon after you see your hens start to squat - it has nothing to do with you rubbing her back, but has everything to do with the fact that she has reached sexual maturity (she's squatting for you because she thinks you are the rooster and she is letting you 'mate' with her). Your SS will start when she is physically ready also. Some pullets will squat for you and some won't (some are more submissive for their humans), but they will all lay just as many eggs whether or not you have a rooster (and whether or not you 'pet' them).
The light is another matter - increasing the light to 16 hours a day can definitely increase the amount of eggs you get from your pullets in the winter.
The light is another matter - increasing the light to 16 hours a day can definitely increase the amount of eggs you get from your pullets in the winter.