Have you checked your squat form? Squats, if done properly, will strengthen the knee.
- When squatting, your knees should not come past your toes. This of course is a rule of thumb and will differ with your body type. If you have really long femurs your knees will go past your toes slightly.
- Try not to let your knees drift in (as you've already said).
- Point your toes out so that they're pointing in the same direction as your knees track.
- Push from the heels, not the toes; this will require flexibility of the hips and hamstrings.
Here is an article on squat form. Don't worry that it talks about squats using a barbell, the sections on squat form, squats and knees, and squat benfits apply to bodyweight squats as well.
And here are some tips on knee health and squats http://stronglifts.com/10-tips-to-stop-your-knees-from-popping-and-cracking/
And finally some exercises to help you strengthen your hips and glutes to help your squat http://stronglifts.com/how-to-optimize-posterior-chain-power-glute-activation/
I should mention that I'm not a physiotherapist or a doctor and I'm just passing along information that I've used in my own training. Train smart, listen to your body and follow your physio's advice.