Right now, I\'m using msysgit to work with my own private repositories stored on a ec2 Amazon Cloud Server using SSH.
Until now, I\'ve been able to successfully connect
Here is a nice write up of what you are exactly looking for.
Edit: I just tested with the following steps and it works like charm!!
1. Click on repositories on the left.
2. Pull your copy of the repo from the remote , using git-shell
3. Drag and drop the Folder on to Git hub for windows dashboard.
4. now double click on the repository dropped on to githubW. It should be listed as a local repository.
5. it says login required to your non-github remote!
Credits
For some wierd reason, GutHub didn't make it straightforward to use non-GitHub Remotes. (As you said, they say that they support non-github remotes though!!)