So, you have a problem with SSL (https) connections. If you're using Internet Exploder, have a look in Tools/Internet Options/Connections/LAN Settings. Check all those boxes are unchecked. If you have mistakenly configured a proxy for HTTP only (and not configured for HTTPS), this behaviour would occur.
Other test - open a command prompt (start/run/cmd) and type:
telnet signin.ebay.com 443
If the Window goes blank, you've just connected on the SSL port and the problem is browser related, not network related. If it hangs and says "timed out", your problem is network related, not browser related.
See how you get on with those for starters....