I see in their documentation where they clearly instruct you to include this import:
import com.google.android.gms.common.GooglePlayServicesClient.ConnectionCallbacks;
However, the error being thrown is expecting a different class than GooglePlayServicesClient.ConnectionCallbacks, it’s asking for GoogleApiClient.ConnectionCallbacks. Try changing your implements to use the more-qualified class name. That looks to be the only possible thing throwing the code for a loop and without the explicit qualified classname, it will default to the directly imported class name.
It’s always tougher when you have to question the manual.
Edit: I mean a change like this:
public class MainActivity
extends ActionBarActivity
implements GoogleApiClient.ConnectionCallbacks,
GoogleApiClient.OnConnectionFailedListener {