Hi Frank,
Without going into a lot of technical background, the short answer is to just always manually add the .qxp extension to your filenames when saving. When you keep them on the Mac drive, they should have additional file metadata that will cause them to be recognized as Quark documents even without those extensions. But it sounds as though your backup device is losing that metadata, in which case you may need the extensions. And if a file has the extension but is not recognizing a double-click, you can always open the file from within the application. (You can also try using Get Info in the Finder on one .qxp file, make sure the Open With pop-up displays QuarkXPress, then click the Change All button to tell the system to always open .qxp files with XPress.)
Hope this helps,
John