Hi,
I have not got Google Hangouts on my iPhone.
I can describe things from the Mac end.
I never signed up for Hangouts as I have been using my Google ID with GoogleTalk for years.
I have never signed up for their 2 step verification but have enabled "Less Secure apps" as iChat and Messages both only use SSL to connect (so does Mail but Google used to ignore that fact as well - different servers)
This meant that Messages used the talk.google.com server it has used since iChat 3 (in iChat 3 there was not a Google entry point and you had to change the server after setting up the account)
More recently ( the last two weeks) I have been trying to reset Messages back to using the talk.google.com server as it caused Google to lock me out via Messages the other day.
I got this email

The Google System seemed to identify my messages Login attempt as a different computer despite the time being my login time.
the Server has changed to gmail.com and is not using SSL and does not seem to want to change.
It would seem that Google have made some changes to the GoogleTalk and Hangout servers - probably aiming for "better" integration that is causing some headaches.
I can only suggest reviewing your settings at Google and on the devices you are using.

9:56 pm Saturday; May 21, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone and an iPad (2)