Update: I was able to get the launcher to open (with much trial, error, and help from EA answers) by performing the following steps - proceed at your own risk, I take no responsibility:
Found and used the ZoneAlarm clean uninstaller tool.
Reinstalled the -previous- version of ZoneAlarm.
And it's working again. Attempting to reinstall the newest version brought the problem back. Reinstalling the old version over the new one didn't fix it, either. There must be something funky in that new update that just doesn't gel with TOR unless completely removed. I will try and contact ZA about this, see if they can patch the issue on their end.