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?
Well, we have to divide the body up into chunks that make sense.
For exercises where we push forward, e.g. the pushup, we train mainly chest, triceps, and front deltoid.
For exercises where we pull backward, e.g. inverted rows, we train the upper back and biceps. These muscles are opposite the ones in the pushing exercises.
For squats, I'm not sure we can follow the same pattern. During the squat you train a lot of quadricep. On the other side of that is the hamstring, but the squat trains that too.
During the squat we also train the lower back. On the other side of that are the abdominal muscles, but again, the squat forces you to engage this muscle group too, in order to keep correct posture.
This is actually why we love and preach the glory of the squat. It does so many things all at once.
Now, while it does train all these muscle groups, it trains some more than others. For instance, the quadriceps are far more active than the hamstrings. Luckily we have other movements that train much of the same areas, but with different foci.
For instance, the deadlift also trains quads, hamstring, lower back, abdominals etc, but it requires more hamstring work than the squat.
So as far as your question goes, I'm not sure the squat has an "opposite" exercise in that regard. Just complementary ones.
Oh, and whenever you're in doubt as to whether you should be doing this move or that move, the answer is usually both. Variety is key.
I'd argue the opposite exercise to a body squat is a Hanging Knee Raise. 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