Yes, you can use the Windows GitHub client and the GitHub Desktop client with GitLab, BitBucket or any other hosted Git solution.
We only use it with HTTPS and you'll need a valid certificate if you do use HTTPS. It may work with HTTP as well. We never did get SSH to work completely right since it's a tough to inject your own SSH keys into the application.
If you want to clone a repository, you have to drag and drop the HTTP URL onto the GitHub application.
I was unable to get the drag and drop trick to work on OS X. But you can add locally cloned repositories into the OSX version and then the application works like normal. And OSX supports SSH keys unlike the Windows version.