Squats are good, although it's always good to include other leg exercises that hit different muscle groups. Lunges, deadlifts, high cleans, side lunges, straight leg deadlfts, etc.
In the context of Super Squats as a whole-body workout, I was going to do a modified version (Bent over row, dumbbell press/bench press, shoulder press, squats); so it shouldn't be as taxing on my body as the full workout or alternate program. Though as I said, if there are any areas in the lower body the squats alone are missing or not hitting as hard as they could be hit, I would be open to add to the workout.
Depends on the person. When I used to do heavy legs, I couldn't do any running for a few days. You may want to take it slow and see how your body reacts.
Edit: I was worried about overtraining my legs due to the running. Would it be as bad if I had to run the day before? eg Run Mon, Squats Tue, Run Fri
I'm confused, is your overall goal focused more on strength gains or participating in martial arts? If martial arts, which one(s)?
Edit2: When I start back MA training, I'll be at most training twice a week and will probably sub another running day, and the training days would be limited to Thu and/or Sun unless I hit striking on Friday, but that one is not a priority for me. These are both just hobbies for me.