Some plastics actually won't melt (or slump) at polymer clay curing temps, but many will. You might just try baking one in a 275 F oven (265 if you're using one of the Fimos), but I'm not sure exactly how you were doing that anyway. Generally, to attach a stud finding to a decorative piece of polymer clay, one would poke a small pad/sheet of coordinating clay all the way down over the pointy rod part only so that it could attach to the decorative part on the front side of the finding, thus entrapping the flat part of the stud with clay and creating a mechanical hold.
If that's what you meant, then enclosing the plastic pad inside the clay like that would further protect it from the heat, and baking might not be a problem... you could also always cover the uncovered rod part of the finding with aluminum foil (or poke it into a pile of cornstarch or baking soda) to help protect it from excess heat too.
Many glues would hold a lightweight bit like this of clay to a plastic pad, and there are special glues for plastic
that you might want to check out, but 2-part epoxy glues are pretty darned strong for most materials. (E-6000/Goop or similar strong silicone based glues would work, and probably even a strong "white glue" --GemTac or Jewel-It are supposed to hold to metal, etc., too).
Polyurethane glues (Gorilla Glue, etc.) are even stronger than 2-pt epoxies, but they do swell a bit while curing (and are yellowish) and need to be clamped or weighted (...can just remove excess glue as it squeezes out though).
For gluing smooth surfaces, it can always help to rough them up a bit (unless using an instant "superglue" which requires exactly matching surfaces). Many glues don't fully "cure" when they first dry too, so be sure and not apply excess pressure for about a week for some of the glues.
You could always use a plastic-coated wire to make your own findings too (the plastic on them is usually fine at clay baking temps, and will actually bond with the clay since its a related material).
Or you could hang your clay bits (preferably with a hole in them) from a thread for "thread earrings"... just use something besides metal chain for the "thread" that you're not allergic to. (You said you can't wear metal at all, right, not just nickel containing metals... many with those problems just buy real silver findings, then switch off the dangle parts if they're the hook type.)
Guess you could always "wrap" the polymer parts onto a stud too, if you did it decoratively, perhaps with decorative thread/string perhaps soaked with glue or only glued at junctures.
There's more about attaching polymer clay to various kinds of findings on this page at my site, if you want to check them out:http://www.glassattic.com/polymer/jewelry.htm
(... click on Earrings especially...)