What IP address do you have for the LAN interface of ZS?
Are you LAN hosts using the IP addresses from the same network?
ZS LAN: 192.168.1.1
LAN host: 192.168.1.2
Having made sure of that, ping ZS LAN IP.
Have you specified the default GW for the LAN hosts to be ZS LAN IP?
What are you using for DNS on the LAN hosts?
You can specify DNS on ZS to be OpenDNS IP addresses, block DNS port in the firewall and use ZS LAN IP for the DNS server.
Please post ipconfig/ifconfig output from the LAN hosts.
And see if you can ping ZS box from the LAN, then if you can ping DNS server.