Those Teflon base bins are great. They have made painting my jigs so much easier. I'd definitely take them out before baking. Here's how I do it. Heat my jigs for a couple seconds in a blow torch, quickly insert the Teflon pin, dip into powder paint, pull pin out, hand in toaster. Once I get the batch done, I bake them all.
Like others have said, a fluid bed helps a great deal with the hook eyes. If you don't have one and don't want to build/buy one, try shaking the jar of powder paint after every couple jigs (with the lid on). This will "fluff up" the paint, aerating it slightly. This should help until you can get yourself a fluid bed. Also, don't heat your jigs too hot, that will also get too much paint on the eye, even with a fluid bed.
cadman, what do you use for the bottom porous part of the fluid bed cups? I built myself a fluid bed, and have a few cups from a retailer with the white porous material, but need a couple more on the cheap.