To all the existing users as well as the new users receiving the error: Unfortunately, an error occurred while connecting to Google, please try again.
This is obviously an issue with the token auth call to Google. Something has changed. Somewhere. I noticed this when trying add a fourth site using this plugin. My three other sites were reporting, but numbers were low. My fourth site would not authenticate, and the manual add of the UA number took, but something was still wrong. Especially with real time viewing at GA.
So I went to one of my other sites, and clicked re-authenticate with Google and what do you know, it failed. This happened on all my sites. So I placed the code in manually, but my numbers were all off. All of a sudden.
The author has yet to chime in on this, and I feel as though this is the case, this is not a high priority for him. OR he just might be too busy.
Anyway, my recommendations for you is to check your theme. Most themes these days (yes, even the free ones) come with some sort of config box, or location where you can put the GA code in your site's header.
You will have to manage your advanced settings directly from GA, but if you want correct and accurate reports, you will have to do a little work for it. At least until the plugin is fixed.
There are numerous articles on how you can accomplish this if your theme doesn't support direct input of your GA code, just look around.