Is it for your back? If your doc has told you twice and the fact that it is naproxen I am going to guess it is your back. Especially if it is chronic you don't want to mess with that. What I know of naproxen and other nonsteroidal antiinflammatories is they are a bit hard on the GI system. There is a reason the instructions are to take with food... can cause really bad belly pain and other GI symptoms like nausea, vomiting, and diarrhea. So rather than be concerned about the drug interfering with the cleanse, it might make you kind of ill (besides detox) to not take food with it. Can you ask your doctor what the effects of waiting a few days might be? If you don't want to tell him about the cleanse you can say you have to drive or operate heavy machinery (like the vacuum?) a lot for a couple of days and you are concerned about being drowsy.
I recently suffered an injury and threw my Vicodin down the toilet after it made me have a dream about seeing rats with X-Ray eyes in my bedroom. I ran outside with only my panties and a cast on and locked myself out. This is a true story and only happened about a month ago! I hope you feel better soon and are able to sleep okay!