It depends on the game and what port it uses. Any decent firewall can block ports, but you need to find out which ports the game is using. If the game uses port 80 (HTTP) or any other common port then your out of luck.
i think the only way to know how many games are running over a network would be if you were the server are apart of the network- say you wanted to make your computer a server for a game network, the software you would be given to make your computer the server would be able to tell you how many games were running on your server- i know a little about how this works maybe you could specify the game and why..
You could use a sniffer and look for the protocol that any specified game was using on the LAN.
If the LAN is a switched enviroment (layer 2 switching) then you would have to use a port monitor on one interface on the switch, and set you NIC to promisceous mode. If the LAN is 1 brodcast domain then you just need to se your NIC to promisceous mode and start sniffing :-)