It took me a bit to figure it out, but I found out how to get a "non-gmail" google account to sign into the google+ app.
You need to be out of the app.
Press the Menu button for your phone, and go into Settings.
Go into "Accounts & Sync"
Press "Add account"
Press "Sign in"
Sign into the google account your Google+ is registered to
Set how you want your sync to be and confirm
After you do this, go back into the Google+ app and your correct email will be listed. Click on it and enjoy!