See
this wikipedia article for a definition/explanation of network gateways.
Right now your router is acting as your gateway to the Internet. Your present configuration is fine
if you are manually configuring your client machines (for example their browsers) to use your proxy. If you want to run a
transparent proxy (which does not require manual configuration of the clients and where the clients have no choice but to use the proxy), then you must set your
squid machine up with two NICs and
force all the traffic to pass through that machine. In which case, that machine will also be acting as a gateway.
To do this (run transparent proxy) you would hook one NIC up to your router and the other NIC up to a hub/switch where you would also connect all of your client machines. The client machines would then be running on a separate subnet from your router and you would probably want the
squid machine to become a DHCP server for that subnet.
EDIT: If you are manually configuring the IP addresses, etc on your client machines, you won't need to set up DHCP