I'm also having the same issue with Gmail not connecting. Email to my own domain server is working without any problem. I tried the checking the boxes for Contacts, Calendars and Notes. It did resolve the issue, however, I don't want Google's Contacts, Calendars and Notes on.
But, if I remember correctly, I had a similar problem with connection issues between Google Mail and Apple Mail app awhile back. The solution was to change the outgoing server address in the Accounts section of Mail's Preferences. (I found the recommended server somewhere in the GoogleMail forums.) But now I see that Apple mail has cleverly gotten rid of the Advanced Settings in Apple Mail. I'm going to refrain from adding a sarcastic comment. But I will look further into this. It is possible that Google mail has once again changed their preferred STMP server.
I should also note that Little Snitch only recently started to report and request connection to: googleapis.com (process from: /System/Library/Frameworks/Accounts.framework/Versions/A/Support/accountsd). I'm not sure if this has contributed to or caused the problem.