What software are you using for your proxy and what method does it use to download webpages? If you are using a proxy that is written in PHP and it utilizes cURL for downloading webpages, it should work with SSL as the HelioHost server has cURL w/ SSL support enabled, and compiled in PHP. I'm not very experienced in whatever CGIProxy uses (I assume Perl? I know only a little Perl), so I'm not sure what methods it supports and and which one CGIProxy utilizes. The problem may very well be a software limitation.
Basically, if you want me to install a package on the server or enable something, please tell me. Otherwise, you'll need to figure that out before you ask me to help, as I really don't know what this situation requires.