Can't log in to Hangouts
So this turned out to be a pretty easy fix, once it dawned on me what it was. 
I just recently installed the Xposed framework, and the XPrivacy module. 
When I uninstalled the old Hangouts, it must have deleted all the old permission settings in  XPrivacy, and then when I reinstalled it, it defaulted to maximum restriction. 

With access to the phone accounts restricted, Hangouts was not able to see my google accounts. 

The fix was as simple as fixing the permission settings for Hangouts in the XPrivacy module.