Today I had a problem with Outlook Anywhere at a customer site - everything was configured correctly however Outlook Anywhere would still not work! The server was Exchange 2007 SP2 running on Windows Server 2003 R2 Standard Edition x64.
When using the Exchange Remote Connectivity Analyzer the following error was experianced.
Testing HTTP Authentication Methods for URL https://webmail.example.com/rpc/rpcproxy.dll.
The HTTP authentication test failed.
Message: The underlying connection was closed: The connection was closed unexpectedly.
1. Disable outlook anywhere via EMC
2. Remove RPC proxy component via PowerShell, Server Manager or Add and Remove Programs (Windows Components) depending on your version of windows. You can also use this command: servermanagercmd -r rpc-over-http-proxy
3. Reboot the server
4. Install RPC proxy component via PowerShell
5. Command: servermanagercmd -i rpc-over-http-proxy
6. Enable outlook anywhere
7. Restart Microsoft active directory Topology service
This should resolve your problem, did for me.