Synchronize google apps with private server?
Yes and no. It's not possible to use your own server, but Google does offer a hosted platform for businesses called Google Apps for Work. It gives you a Gmail interface (both the webapp and the Android app) and Google accounts in your own domain (which you can register through Google or independently), along with a private Google Drive for your domain and some other of their apps.

All the data are still stored on Google's servers, but because it's an enterprise service, you have service level agreements, non-disclosure and confidentiality.