When re-installing Mountain Lion from the Recovery Partition, the installer needs to check installation eligibility with Apple's servers. If your computer needs to access the Internet through a proxy server for whatever reason, the installer won't pick up on this; it will attempt to make a direct connection, fail, and tell you to contact AppleCare.

As per my earlier hint (10.7: Get the Lion installer to work behind a proxy server), you could simply use the networksetup command in Terminal to get around this. However, as of 10.8 Apple's software download servers appear to require additional checks to verify machine eligibility which - if you're behind a squid proxy server - may require additional configuration changes to squid itself in order for it to work. You may therefore need help from your network administrator for that part.

These are the rules I added to our squid.conf file; I leave them commented out in our configuration unless I need to reinstall Mountain Lion.

# Uncomment to reinstall Mountain Lion
forwarded_for transparent
via off
visible_hostname proxy.local
httpd_suppress_version_string on

Remember to reconfigure squid in the normal way after saving your changes.

The rest of the instructions are the same as they were in 10.7: start by booting your computer into Recovery Mode, and access Terminal from within the Installer's Utilities menu.

The networksetup command can configure a vast array of network options, a full list of which you can see by typing networksetup --help. The two options we're concerned with for this purpose are -setwebproxy and -setsecurewebproxy. You need to enter a service name with each command: in most cases this will be Ethernet for wired connections, and Wi-Fi for wireless ones.

So putting this all together, to tell the Installer to connect using the proxy server proxy.example.com via Wi-Fi on port 8080, we would type:

networksetup -setwebproxy Wi-Fi proxy.example.com 8080
networksetup -setsecurewebproxy Wi-Fi proxy.example.com 8080

Substitute your own values, of course. Quit Terminal, go through the Installer steps again and you should be good to go.