Hi, Alex. Thanks for the reply.
I've disabled SEF on the website and turned on Debug Mode in the JFBConnect component, but I'm still not getting any error messages and still not being logged in.
However, updating Joomla to the latest version and updating JFBConnect to the current v8.0.1 seems to have worked on one of our test sites. The other site, sadly, still has this Google+ error/issue.