I fixed the problem and wanted to share it with those of you on this thread so you won't have to deal with it in case this happens to you. I installed a caching plugin last year called Super Cache. It basically caches the whole site regularly in an effort to speed up the response time on the site. It created a directory OUTSIDE of the regular WP directories. When I uninstalled it (because it sucked) it left the folder with all of the current cache as a remnant on my server.
For some unknown reason it was defaulting non-registered users to that folder. I must have tweaked some plugin or WP setting that triggered it. Lesson learned, stay away from this plugin.
Thank you all again for your inputs and help. I really appreciate it.