Although our libraries don't currently include an HTTPS client, we are most likely adding it soon - we had to write one to make it possible for our forthcoming EWS library to work on Windows CE because we soon found out that .NET CF's HttpWebClient suffers from at least two serious SSL-related bugs that make it unusable. The lack of support of TLS 1.1 and 1.2 is also becoming an issue.
Because several customers expressed interest in a HttpWebRequest replacement object for .NET Compact Framework, we created a proof-of-concept library that is available from Rebex Labs: http://labs.rebex.net/HTTPS/
Unfortunately, SHA-2 is only supported on Windows CE 6.0 and some Windows Mobile 6.5 devices.
Update: As of April 2016, we support SHA-2 on all Windows CE / Windows Mobile 6.x devices, but if it's not supported natively, you might have to provide a custom certificate verifier to make SHA-2 based certificate validation work. You can find more information about this on Rebex HTTPS page.
To give it a try, the three DLLs from your application, add "using Rebex.Web" and "using Rebex.Net" and do the following once when the application starts:
var creator = new HttpRequestCreator();
// enable TLS 1.0, 1.1 and 1.2 (also supports legacy SSL 3.0)
creator.Settings.SslAllowedVersions = TlsVersion.TLS10 | TlsVersion.TLS11 | TlsVersion.TLS12;
// register HTTP request creator to replace .NET CF's default HttpWebRequest
Then try running your code. If it doesn't work, please let us know. This is an experimental code that has not been tested to handle all possible scenarios yet.