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: 
What is the opposite of sit to stand?

Break the motion down concentric vs eccentric contractions, joint by joint -- quite simply there isn't one.

A great exercise to add to your arsenal is the hip dominant Split Stance Romanian Deadlift

<https://www.youtube.com/watch?v=QDEMmKocxbM> (Direct Example)
<https://www.youtube.com/watch?v=XowKMitOVNc> (This Guy Knows His Stuff)
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