Question: Is there a good bodyweight antagonist exercise to the squat? In the same way the push up has the pull up or the inverted row?

Answer: 
If you have a hanging bar and some inversion boots you can do the bodyweight squat upside down ie hanging from feet "lifting" glutes to heels and back down
You could hang from a lat pulldown machine and fix your feet into place. This is the opposite of the squat. the hip flexors are the main muscle that is working i think. you could also do leg raises of some kind. I am not sure why you would want to do them tho