I'm having this exact same problem!! I can't log into my main account in google chrome using google plus login and yet logging in with an e-mail address seems to work fine in the SAME browser on the same computer.

Two things are happening:

1. Problem similar to what you're describing here- click log in and game starts to load but then immediately says check connection, or game down for update, or you have been disconnected from server.

2. A different problem where the "login" button for google plus login becomes unclickable. I can log in just fine with google plus on tablet, so there is definitely something in chrome preventing me from logging in using google plus.

If anyone can shed any light this will be much appreciated

*hyperventilates*