So far it looks like some issues with SSL/TLS in Mono/Xamarin.
https://bugzilla.xamarin.com/show_bug.cgi?id=42805
After all the failing code just sends a POST request out via an HttpWebRequest instance and lets the system do all the SSL stuff. And then it fails.
As a workaround try to use a client channel described here: HttpClient-based Client Channel for Xamarin platforms (and other platforms supported via PCL)
Seems these parts of Mono should use newer SSL code paths