Consider java applet loaded in browser.
Since java uses its own virtual environment, it will be able to show local Ip address to the clients.
Good way to start i guess -> http://reglos.de/myaddress/MyAddress.html
And for Java to obtain IP: How can i check System IP Address/Host Name in Java?
This way you wont have to change current environment. Otherwise it will probably involve squid reconfiguration.
From the comments you have posted i can see that the purpose of presenting the IP to the client is to make them read it to you and allow you to connect remotely. You have also mentioned that running an app on a client side is an option. This immediately pushed me towards BGInfo http://technet.microsoft.com/en-us/sysinternals/bb897557.aspx - especially if the clients operating system is Windows.
The easy Java aplet to put on a page:
function myIP(){ var vi="uses java to get the users local ip number"
var yip2=java.net.InetAddress.getLocalHost();
var yip=yip2.getHostAddress();
return yip;
}//end myIP
alert("your machine's local network IP is "+ myIP())