I've had another #1 page rank in google from someone looking up why cider/apple juice gives them the shits. It's nice to be recognized folks, but I don't have the answer. My site pops up because 'cider' and 'apple juice' are all over the place, and on the general
cidermaking page there is a mention of "Uncle Eddie's 'give-you-the-shits' cider". Now I drink a lot of cider and a bit of apple juice (aka sweet cider), and I don't get said gastrointestinal problems. Maybe it's because I'm used to it, maybe I just make good cider.
I could venture a guess as to why you're seeing the gravy train after drinking cider/juice, particularly the sweet, unfermented variety. There's a lot of readily fermentable sugars in juice, so maybe juice combined with your gut flora is creating an in-body fermentation and your butt is acting as an airlock/blowoff tube. Maybe it's the large shot of soluble fiber which is going through your intestines like a pipe roto-rooter. These are my two guesses. I'd google it, but it will only refer me back to myself, the google cider shit expert. Again, I don't have that problem.
If you stumble across this page in a search engine looking to resolve a personal problem, I wish you the best, but can't offer much advice except to keep drinking cider so that your body gets used to it. With that said, everything is best in moderation, as you may just be overpowering your bowels with too much good stuff. Good luck on your further search, keep a clear path to the loo, and don't knock cider in general. It's good stuff and good for you.
TB