Change the name of 'CraftSessionId' from HTTP header requests? I am running a Nginx server and tried to hide the 'Set-Cookie' parameter from the HTTP header output when using curl -I hostname.com - I applied the Nginx configuration using: 

proxy_hide_header Set-Cookie;

Using this worked as intended - until I noticed that any request to login to the control panel simply resulted in reloading the admin login page.

Is there a way to change or hide this that still allows users to login to the control panel?
I'm not sure I'm understanding 100% what you're trying to do, but by preventing the Set-Cookie header, aren't you essentially blocking any cookies from being set in your browser?

All of Craft's authentication is cookie based, so it's not surprising that trying to login to the control panel doesn't work.