Is it possible to use OSX to publish website changes via the command line (not through the git method)?
If not, then how would a UNIX based build server do continuous integration deployments for azure websites?
Yes, I believe you can still do this even if you're avoiding git (why?), because you can use FTP.
I think you need the 'Windows Azure command-line tools for Mac and Linux'
"This tool provides functionality for creating, deploying, and managing virtual machines and websites from Mac and Linux desktops. This functionality is similar to that provided by the Windows PowerShell cmdlets that are installed with the Windows Azure SDKs for .NET, Node.JS, and PHP."