Yes, you can re-install the same version, or future versions without problems. Joomla has issues when you rollback to previous versions and then try to upgrade again. Part of that is due to how JFBConnect works (we don't uninstall the database tables when you uninstall, so that you don't loose your configuration settings). However, that causes problems since after you upgrade, the changes have been made, and SQL throws errors for trying re-create the same settings.
As for your original issue when upgrading, We haven't heard of any installation problems like that yet with the new release. If you are using JLinked, you'll need to make sure you upgrade that to v1.2 as well, as there are dependencies between JFBConnect and JLinked now so that JFBC v4.2 only works with JLinked v1.2 or higher. I don't think that's your issue though.
If you do try to re-install again, set Error Reporting to 'maximum' in the Global Config area. That should output an error on white-pages that we can use to diagnose the problem. Other ideas would be to disable all JFBCProfile plugins before you upgrade the component, though that shouldn't be necessary.
Hope that helps. If you have a development site setup, that'd be the best way to test, as a database by itself doesn't tell us everything.
Thanks,
Alex