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
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)