I'll assume you mean the ones that mount it to the door skin, yes they have inserts and yes they universally sieze to the phillips head screws that hold the mirrors on. upside is yours spin instead of just stripping the heads on the screws.  the best solution is the same however.  find a nice drill bit and drill the heads of the screws and with a die grinder remove the inserts. you can get them at any nut and bolt supply place and some creativity with a long bolt washer and a nut will let you crush them without the special tool

Wow thanks.

I had thought they might be Rivnuts that had broken but wasn't sure, especially since both screws acted exactly the same.

I had also suspected I might have to drill out the screw heads to get them out but was hoping I didn't.