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: 
I'd argue the opposite exercise to a body squat is a [Hanging Knee Raise](https://barbend.com/benefits-hanging-knee-raises/). While the squat relies on eccentric contraction of the quadriceps going down into the squat, and then concentric contraction on the way up, the knee raise reverses this with concentric contraction of the quadriceps on the way up and eccentric on the way down.

They're not exactly equivalent, of course. The hanging knee raise works shoulder mobility in a different way, and the bodyweight squat does different things for opening up your hips and ankles, but I think it's still a pretty good opposition.
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)