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).
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)