We're not orthopedic surgeons, unfortunately. From my own experience with a bad knee, this sounds more like either tendon or cartilage problem rather than muscular, if it is in fact in the knee. That's a joint, not a muscle. If you hurt muscle doing squats, it would most likely be in the glutes or the thighs or possibly the periformis, and it would probably just be sore, not injured unless what you don't say is you were doing this exercise with heavy weights. Were you doing squats or lunges? At any rate, if the pain is lingering, you're going to have to get it looked at and probably get some pictures taken. It might be the meniscus, it might be patella tendonitis, it might be just a strain. When I hurt my knee the first time, it happened running, and I gotta tell you, I still haven't gotten a definite diagnosis of it, it can be tricky. I've been told it was a sprain, tendonitis, a torn meniscus, and shoulder shrugs. For me, a patella knee brace and physical therapy and never running again (ugh) fixed it pretty much, But I was older and had a long history of heavy exercise, and you have the advantage of being young -- so be wary of being told you need surgery. If you are, get a second opinion, because if you're able to walk as well as you are you probably don't need it. It all depends on where the problem is found to be. This is all based on your description of the location of the pain -- if it's in the back of the knee originally and now it's in the knee area, again, that's not a muscle area, it's a joint area. But again, there are no orthopedists on here. My guess? You're headed to physical therapy.
There is something called 'jumper's knee" and it is a tendon issue with the patella tendon. You'd feel this just below the knee cap or at the bottom of it. That's tendonitis of the area. You could also have something called chondromalacia patella. This is caused when cartilage is thin behind the knee cap and is caused by muscle tightness, weakness and overuse of muscle. Do you feel any grinding with it too? There is also something called 'runner's knee' (which you don't have to run to get) that my son has had. It's a problem with how the patella moves. This is going to feel more like an ache type pain in the front of the knee. Muscle weakness again can be involved to cause this as well as tightness or even just how the leg, knee move based on foot position (think squats).
I'd personally make sure you take some ibuprofen to help reduce inflammation, ice it regularly, try some arnica cream, etc. Stretching and strengthening if you are able. But it would be best if you could see somebody to get it looked at. Pain after that long of hurting it isn't a good thing. No, I wouldn't worry about nerve damage. But you want to get this worked out. And if they send you to a physical therapist, they are really helpful! They'll get you back up and running (or squatting) before you know it! Let us know how you are doing!
I was doing a similar exercise and ended up straining my lower back muscles and pinched my sciatic nerve. I didn't step foot in a gym for over 2 months. Take lots of time off. your body needs to heal, and doing those rapid squats was a really dumb idea! It was probably the quick movements that did you!