It's a load of b*llocks, really. Sure, zinc is not organic. Zinc fingers, however, are very much so. (See http://en.wikipedia.org/wiki/Zinc_finger for a rare case of Wikipedia being quite useful and accurate.)
You cannot equate organic with living, and inorganic with non-living. A bottle of alcohol is very much organic, but not alive, unless you drink half of it and get into a funny mood.
Of course, there are dead parts of the body, like hair and nails, but then again, that is nearly pure protein (keratin, to be precise), which is of course an organic compund.