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: 
For bodyweight exercises, perhaps you're looking for something along the lines of a **slick floor bridge curl**? [This video](https://youtu.be/ZA8GzhFh_CQ?t=2m26s) demonstrates the exercise. With a little more equipment you can do the **Nordic Curl** [as demonstrated here](https://youtu.be/DQQleh4xUjU).
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