Chiming in - this most likely became my problem too on Touch HD PB631. Sync was working until yesterday relying on the authorization I had since ever, then I tried to logout, and can't login back. I get the blank login screen as well. The PB browser has no problem in displaying
www.dropbox.com, but shows a blank page for
www.dropbox.com/login.
Reminds me of the issue of
changed overdrive authorization. Authorization schemes change and PB is not investing in upgrading old models firmware to keep support.
I found that there is a configuration in /mnt/ext1/system/config/dropbox/dropbox.cfg, but if the scheme changed, I doubt that any other API could fill in the require tokens in order to make it work again. Or?