It's true that Acne people should avoid all sugars, BUT the benefits of the flush far outweigh the temporary problems due to taking in fruit juices. Expect to have a bit of a breakout as your liver releases toxins, too.
Instead of apple juice, you have non-sugar options like Liquid Phosfood, Malic Acid (the active ingredient in apples) and Gold Coin Grass. Consider taking one or two of them to soften the stones and skip the juice altogether.