Ok, here's what I found:
I use a shared SSL server and this program doesn't like the path that is normally used for all my admin functions:
https://www.sharedsslserver.com/mydomain.com/somescript.php
It will load the main backup screen within admin (https://www.sharedsslserver.com/mydomain.com/myadmin/backup_zc.php), but when I actually run a backup, it will return with a white screen and this in the URL bar:
https://www.sharedsslserver.com/backup_zc.php
Clearly, a broken path definition somewhere - but I can't fix it.
When I disable all SSL for admin side, the backup program works for the most part. I can do backups of all files and directories, downloaded or not. SQL backups do not work if I choose to download, or they seem to work if stored on the server, but the files are considered "invalid" when I try to open them.
I guess it doesn't matter, though, as I will NOT be running my store without full SSL on the admin side, so unless I missed something or a patch is written to allow for relative paths, I won't be able to use this backup program.