The Jakarta HttpClient library is very handy for building HTTP clients of all kinds. Its user guide has pages on authentication and SSL.
If for some reason (which one?) you don't want to use 3rd party libraries, the standard java.net.URLConnection class also supports HTTPS, and here you'll find a description of how to do basic authentication with it.
An authenticator is not generally needed for HTTPS connections. It may be required if the server uses authentication, though. If one is required but not provided you should get an error message to that effect.
Whether or not a proxy is needed depends on your network setup. It, too, is not generally required by HTTP or HTTPS, irrespective of whether or not authentication is used.