Third party means not native to Adobe Flash or Flex. So for instance SWF built with Flypaper, Lectora or any other software other then Adobes own software.
Another issue could also be any older SWFs that pull data that were built with 3.0 SDK of Flex.
You say almost immediately, so if your resource area is open and an incompatible SWF file is selected, this could cause the issues. It would so it on any platform, but if you happen to have it open on the windows box. I don't know, I am just throwing ideas out there as the system works fine on both platforms without crashing issues and has since the latest release. It has to do something with either your account or your machines. And since it is happening to Two machines, my thought would be to start with the Studio and the potential issues with your resources.