After playing with the trailing slashes... the combination that finally worked is detailed hereinafter:
Running version: Akeeba Backup Professional rev59EAB5F (2014-02-17)
>> Part size for split archives: 49.99
Username: box.com_username
Password: box.com_password
WebDAV base URL: https://dav.box.com/dav/ (note the last forward slash)
Directory: customername/domain.com/backups (note there's not an starting trailing slash + No final trailing slash, too)
BTW, WebDAV descriptions are missing (see attached picture)
Hope this helps,
Rgrds,