Hi There
Firstly, you need to understand difference between fire-walled connection and proxy.
Your firewall can also be your proxy server, it looks like that is the case on your network.
On the firewall (default gateway) you need to either block direct port 80 ( browsing ) and force your browsers to put in Proxy setting for Internet access.
Or you can use IPTABLES to create Transparent proxy, which will automatically put all browsing through the proxy without users settings.
On the Firewall (gateway) You must add the following rul:
iptables -t nat -A PREROUTING -i $INTIF -p TCP --dport 80 -j REDIRECT --to-port 3128
This will automatically redirect all browsing to your Squid proxy server.
On the Squid configuration, you need to add the configuration for transparent proxy :
http_port 3128 transparent
http_port 3129 intercept
Hope this help