I think trying to add HTTPS support into the arduino is not going to be possible due to te size and complexity of the SSL libraries. What you could do is make the requests to the regular server and use a PHP script to proxy the request to the HTTPS server.
OK, had a quick google and test, found a script at PHP Proxy Script for cross-domain requests ยท Abdul Qabiz that you stick on your not https server, you give it a url parameter that includes https and it returns the results. The URL parameter must be encoded so that the parameters are passed to their ultimate destination.
Its not going to return full pages and images, just the text or html output from the request you submitted, which is probably what you are looking for.