A Hindu Squat is a body weight exercise that emphasizes the quadriceps. It is designed to not only target the quads, but much of the lower body and back as well and is touted by many body weight enthusiasts as one of the best all-around exercises you can do. Here's a video:
I learned of this exercise through Ross Enamait's Underground Guide To Warrior Fitness (I believe the 'second edition' is Never Gymless) and Matt Furey's Combat Conditioning.
As a body weight enthusiast myself I have performed this exercise many times and over the years have come to question its safety. The exercise is to be performed many times in a row; both Ross and Matt state that you should be able to do over 500.
While it is undoubtedly a great workout, through training in yoga, pilates, martial arts, weight lifting, general cardio and other channels I have repeatedly been told that your knee should never or rarely extend past the front of your toes because it puts a great amount of pressure on the knee joints.
I have never heard of anyone encountering an injury as a result of doing full hindu (bodyweight) squats, but I've removed it from my routine in favor of 'half' or regular squats (Mike Diebler in the photo).
Hindu squats will give you a full body workout, but have the potential to cause some damage. While I have found no conclusive evidence of serious injury caused by this exercise, to me it is simply not worth the risk. There are many exercises that can replace the full squat-to-your-heels motion such as the above mentioned 'half'-squat, lunges and calf raises. I recommend you try these instead. I also recommend varying your workout for different ways of doing the same exercise (see this post about fast reps, slow reps and static contraction). Some of the worst injuries people will tell you about are the ones that creep up on them over a long period of time such as joint injuries (bad knees, shoulders and elbows from doing something too much or improperly).
If anyone has any more insight on this I'd be glad to hear about it. If you're a kinesiologist or even a certified trainer or instructor I'd especially love to hear from you.