I managed to figure this out, although I don't understand the reason behind the issue.
Somehow, when I was connected to my home WiFi network Boxcryptor was unable to function properly,this is why for a week it was hanging at the Logging In stage. When I turned off WiFi I was able to access Boxcryptor properly and the Dropbox sign-in worked within Boxcryptor.
Strangely, it continues to fail using my WiFi. I have restarted the WiFi router and everything else works fine over WiFi including accessing the Dropbox site directly. If I connect to a VPN over my WiFi then Boxcryptor also works properly so it's obviously something todo with my WiFi.
I have no special filters or protections that I'm aware of but maybe my ISP has restricted access somehow? I am able to connect to the Boxcryptor site over WiFi, however, so I'm quite confused...