Personally, I’d say that squatting is to sitting what sitting is to laying down. The angle is different, the position is different, and the muscles are different.
As for “is the laying on the back position more comfortable for childbirth,” let me quote from the site I just linked to:
Up to the late 1960’s, the standard position for childbirth was the lithotomy position, where the mother lie flat on her back with her feet tied to and held up by stirrups.
The lithotomy position was an ergonomic nightmare for both mother and baby. No other position could be worse than this position for childbirth. It was said that the lithotomy position originated from a command by French king Louis XIV, who wanted to see one of his mistresses giving birth.
Women hated this position because they could not exercise any control over the birth of their baby. On her back and with her legs up in the air, she cannot move much. The pressure of the baby’s head on her back and tailbone can be very painful.
Also the poor mother has to push her baby ‘uphill’ against the force of gravity. This requires her to strain and push harder and longer. Additionally, in this position, the size of her birth canal can be reduced by up to 30%. As a result, during delivery, the baby is used as a “wedge” to force open the narrowed birth canal.
This causes great pain and distress for the mother, and prolongs her labor. Despite valiant efforts, many women – with the laws of physics and biology stacked against them – become exhausted and could not give birth easily.