Google Hangouts gives "Can't sign in because we couldn't reach Google. Try again" error
Re-installing the 'Google Play Services' updates solved the problem for me.

Simply go to Manage Apps &rarr; Google Play Services and press Uninstall Updates. Now if you start Google Hangouts again it will ask you to update Google Play Services. If you do so, it will bring up Google Play where you can install the update.

After this, Google Hangouts started without any problems on my tablet.