In all honesty, I have heard enough horror stories to sympathize with your daughter's distrust of prescription medications. To extend that to something like a simple zinc supplement, though, is further than I would be willing to go. It's too bad that she lumps those all together.
This is just my personal opinion, but I've experienced great improvement in my mental and physical health from getting better nutrition, including supplements.
Sometimes during those teenage years, 'logic' can take some very strange turns. And unhealthy idealism and going to extremes seem to be part of the package, too, unfortunately. If I met my own teenage self today, I don't think I would be able to convince her of anything at all!