While not a lawyer, based on my layperson's understanding, I agree with the above assessments. There's no problem using any juice you can get your hands on, but you will have to be very careful if you want to use anyone else's trademark in any way. You'd probably be technically safe with something like "Kewlio's Cider, made with Mott's Apple Juice[*]" plus a footnote that explains that Mott's is not involved with the cider fermentation. However, in the trademark realm, "technically correct" is only a defense if you have the funding to defend a very expensive lawsuit from a massive company, so in practice you don't want to go anywhere near a big trademark.
My question is why would you want to use Mott's juice? For a serious commercial operation, you're probably going to want to find someone who'll cut you a good deal on wholesale juice. I don't know much about the juice industry, but I think you'd want to look into the distribution side of things---the wholesale market may look a lot different from what a consumer sees on the supermarket shelves. Certainly I'd look into that before picking out any particular manufacturer.