I use a medicine ball in my home gym to train core body strength. Can't really judge from pictures and video but the Bosu Ball looks a bit gimmicky - maybe not. At any rate you can do much of the same with a medicine ball with the added benifit of being able to use it as a medicine ball, i.e. two person exercises, "wall ball," etc. A 26 pound ball is adequate for most (and I would start with a lighter ball and work my way up). For squats you'd probably want a larger ball.
I use them at my gym for many different things. SteveBJJ is correct that they are great for core. Along with balance training, strength training, and cardio-respiratory endurance training. Working on an unstable surface ups he difficulty of any exercise by a bunch. And it allows for you to hit all of the little stabilizer muscles.