EDIT (8-23): For clarity
I'm looking for an alternative to using web proxy scripts (Glype, PHProxy, etc.) by using proxy software like Squid or Privoxy. I want this new method to operate the same way as a web proxy script, where the user types in a URL into a form, my server fetches the site, and sends it to the user.
What I had in mind was to let the user enter the URL, and have my server somehow connect the user to the URL via proxy software running on my server. So it's as if the user configured a browser to use my server IP as a proxy.
Here's a summary:
Is it possible for users to connect to sites through Privoxy or Squid without having to configure a browser to use the server IP? They first connect to my webserver running on the same server. I don't have control over the users' networks.