I did a low fat diet in college, my hair fell out by the handful in the shower.
All fats are not created equal. Some fats aren't great for you, some are very beneficial, you just can't treat all fats the same. We all need healthy fats in our diet to keep us healthy.
General rule of thumb (just a general guideline):
good fats - fats from plant sources (olive oil, avocado, nuts/nut butters), fat from salmon and other coldwater fish. I try to make sure that 20-25% of my daily calories every day come from healthy fats - I limit saturated and trans fats. Fitday is a great way for me to track this).
bad fats - saturated fats from animal products (good idea to eat low fat dairy products), meat
extra bad fats - trans fat, man-made fats like hydrogenated oil (found in almost all packaged baked goods and margarine)
heart healthy fats
more info on healthy fats from the Mayo Clinic
As far as the unlimited amount of apple juice - weight gain/loss is inherently based on calories in vs. calories out. If you consume 3500 more calories than it takes to maintain your body, you will gain one lb. It doesn't matter how much fat is in apple juice, if you drink 7000 calories of apple juice over what it takes to maintain, you will gain 2 lbs.
If you are interested in a low sugar diet, I guess that would be Sugarbusters (I think there's a forum for people on Sugarbusters).