I don't know whether this will work for everyone, but a wonderful Apple tech support person puzzled out that my .backup files could be read by changing the extension to .dmg
That was it. Whew!!! I'd been trying for weeks to recover my information.
Backup was part of the Macintosh OSX system before Time Machine. Backup is no longer supported under OSX Lion, or even Snow Leopard, and no one seemed to know any way to get to the files. The Backup program is still on my computer, but it was confused and wouldn't open the files.
BUT!!! Changing the extension from .backup to .dmg was all it took to have my files open up in the CopyWrite application in which they were created.
Thank you, Matthew at Apple Tech Support!!!!