Hi Sandy,
With your chrome configuration is not possible to hide completely the web part using the ezEdit options. I googled it a bit for a solution and I found some javascript code that'd do that. You still need to code a bit to show/hide based on user groups. I'd suggest to use ezEdit options to hide fields and add the javascript to hide the whole webpart.
If you accomplish that please share the solution. The community would appreciate that.
Please follow the link:
http://wyly.wordpress.com/2009/05/22/showhide-multiple-web-parts/
Thanks,