And what about Chrome, Opera, and Safari? (IE can go pound sand)
Clarifying, I have the power to reconfigure Squid if necessary.
If you are asking if Squid can proxy SSL connections, then yes it can. It cannot cache SSL connections for obvious reasons.
If you are asking if you can do HTTP proxy (or caching) via a Squid proxy that requires HTTPS, then yes it can if you aren't doing transparent proxying. It is the same as using HTTP proxy URLs. Of course, Squid has to be set up for caching via SSL.
This is useful if you run a cyber-cafe or open Wi-Fi and don't want everyone's traffic to be visible to everyone else.
See the documentation for https_port: http://www.squid-cache.org/Doc/config/https_port/
You can also use SSH tunnel.
You also can run any proxy (like squid) at your-proxy-host bound to 127.0.0.1 and use
For the latter you can also substitute ssh with socat or stunnel.
Firefox can't talk HTTPS to proxy yet ( bug https://bugzilla.mozilla.org/show_bug.cgi?id=378637)
Chrome can do that: http://www.chromium.org/developers/design-documents/secure-web-proxy