What you are looking for is Transparent proxying (
http://tldp.org/HOWTO/TransparentProxy.html), however for this to work all the traffic has to go through the squid machine first.
You could look into making the linux box the gateway and it sends all traffic to the router. In my setup, my linux box also does DNS caching/DHCP serving so the router effectively becomes an ethernet modem.
At least this is how I've set mine up and it works well.
Hope this helps.