To answer your question, your biggest concern is the APAP (Tylenol, acetominophen) with AS. I would never take even one Tylenol while juicing.
Try to get your doc to write you for single entity hydrocodone or oxycodone. Problem is that these are Schedule II drugs versus Schedule III for fixed combination hydrocodone. Docs prefer Schdeule IIIs becuase they can be called into the pharmacy (in most states) and don't get extra DEA scrutiny. Doesn't mean he won't write the CII, they do it all the time for Percocet, Oxycontin etc.
Downside is that as soon as a patient asks for single entity narcotic it screams drug seeking behavior and alarms start going off in the doc's head. All that stuff has a good street value. Might just want to tell him you are juicing and don't want the APAP.
I would stay away from the Oxycontin unless you have serious chronic pain. Otherwise, stick to the immediate release stuff and use it sparingly. Less chance of getting hooked up bad. Good luck. Pain sux.