How to ping eth0 from eth1 on the linux machine

If you don't need a transparent proxy, then you don't need to bother giving eth1 an address. All the traffic can be handled by eth0.

If you do need a transparent proxy, you have two options here.

Option 1: Bridge it

Configure both Ethernet faces into a bridge group, and then use iptables to handle the intercept. You assign the virtual bridge interface an IP address.

Option 2: Route it

Configure your Squid server as a router. Install dhcp on it to ensure the rest of your network can get addresses. Then use iptables to handle the intercept.

The simplest solution is to disable reverse path filtering. By default linux filters out packets coming in on an interface which it thinks should have come in on a different interface (because the packet matches the other interface's subnet).

To do this

echo 'net.ipv4.conf.eth0.rp_filter = 0' >> /etc/sysctl.conf
echo 'net.ipv4.conf.eth1.rp_filter = 0' >> /etc/sysctl.conf
echo 'net.ipv4.conf.lo.rp_filter = 0' >> /etc/sysctl.conf
sysctl -p

This adds the setting to the sysctl config file and then reloads the config. You can also temporarily disable the setting by doing echo 0 > /proc/sys/net/ipv4/conf/eth0/rp_filter.