Hi Nate,
What you're looking at is called endpoint protection, and is a booming business. It only allows endpoints to come into the LAN which are 'trusted' (e.g. correct AV scanner and patch level), and otherwise redirects them to a quarantaine zone where if you want so they can get the right AV level.
The usual approach involves relatively expensive black boxes, either Cisco gear (their protocol being called NAT) or 802.1x compatible gear. 802.1x comes from the WiFi world but is useful in a broader sense.
In your (budget) case I'd go for a cheaper, but very cleverly designed, route using the DHCP servers. The main vendor I know of is Sygate (now merging with Symantec), and the tool is the 'LAN Enforcer' which is part of Sygate Enterprise Protection.
LAN Enforcer sits in front of the standard DHCP server that you have. When an endpoint asks for a DHCP lease, it first gets one in the quarantaine zone from Enforcer. Then Enforcer has the endpoint checked (AV level etc.) because it should have the SEP agent installed which does this for you. If 'red flag', endpoint remains in quarantaine zone forever. If 'green flag', then Enforcer contacts the real DHCP server and asks for a lease which is then passed on to the endpoint. Now only it can work fully in the school LAN.
Note that LAN Enforcer/SEP will cost you money, but contrary to the hardware-based models it does not need any changes in your black boxes.
Regards,
- The Netherlands