Brian Davidson kindly sent in the following note on how he fixed his problem with offline files settings resetting themselves after a reboot:

I've been struggling with this issue for awhile and I noticed you had a post about it with other users responding having had the same issue. Nothing I've seen anywhere solved the issue. What I discovered that fixed the problem was that even when the Welcome Screen was turned off, Fast User Switching was still being seen as on for whatever reason. The fix: make sure no other clients or login replacements are turned on, goto the User Administration screen, turn on Use Welcome Screen, then uncheck Use fast user switching, then turn off Use Welcome Screen. Apply changes, enable Offline files and apply, reboot and it should stay checked. You should then be able to reinstall any clients (Novell for example) that you were using before. Hope this helps.

Update: In the comments below, Joyce Hettenbach found an additional step that might be needed. This has also solved things for at least two other readers of this blog.

Finally, I found the answer to the problem. This hint sent me in the right direction. Even though the Welcome Screen and Fast User switching were OFF/DISABLED the all important registry key had not been changed. The following key was still set to "1" when I changed it to "0" offline files stopped unchecking itself after every reboot.
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Winlogon]

Don't ask me why turning off the Welcome Screen and Fast User Switching was not setting this key to "0", but that is exactly what the problem was. Could it be some bug in the Fast User switching interface? When the box is unchecked in the GUI this key does not appear to get changed...

Edit: I've turned off comments on this thread - if you have a question, please go here instead.