Can I use SourceAnywhere for VSS on a Windows machine to check in/check out and deploy files to a Linux server?

Yes. SourceAnywhere for VSS provides a Web Deploy feature which allows you to deploy files to a web server.

For more information on Web Deploy, please refer to: http://www.dynamsoft.com/help/SAW/CO_Deploy.htm

To check in / check out files from a Windows machine to a Linux server, you can use SAMBA to integrate Linux into the Windows network. For information, please see: http://www.reallylinux.com/docs/sambaserver.shtml
 
Also, SourceAnywhere for VSS has Java Client that can work well on Linux.See: http://www.dynamsoft.com/help/SAW/CJ_Overview.htm