I fasted a few times in the 70's, when it was kind of an "in" thing to do. I did a 3-day water fast (complete with enemas, which are supposedly necessary when you are not taking in any fiber) and a couple of juice fasts. I juice fasted for about 3 days when I had some wisdom teeth removed, then went on to a raw foods diet for a couple of days afterwards. The dentist said he was surprised at how quickly I was healing (I didn't tell him about the fast, as he had told me to start eating as soon as I could).
I would definitely NOT recommend doing any kind of long juice fast on your own (21 days would be long). With long fasts, it's important to transition back into regular eating gradually so that you don't get ill (you start back with raw foods and more easily digested fruits and veggies, or broth, then go to starches, then proteins and fats). Also, an extended juice fast would have a tendency to have the same effect on your body as prolonged dieting with a very low calorie diet, that is, it would reduce your metabolism and make it easier to gain weight afterward. You also have to be careful not to have too much fruit juice, as they are high in sugar (natural as it may be).
If you want to try a juice fast, try a one-day fast first. Or you could do a weekend cleansing diet of raw foods. I think that eatiing "lighter" once in a while, and maybe taking some herbal teas that are supposed to be good for overall health (reishi mushroom, astragalus, others that have antioxidants in) can be as beneficial as fasting is supposed to be, and doesn't seem as apt to mess up your metabolism.