Allow very limited Internet access

Clarifications

Checkout untangle.com. This is a open source security server which can be placed on a stand alone machine (which I prefer myself) or in "rerouter mode". This has solved many issues for my clients and provide another layer of protection. I take old boxes (P4 2.00 Ghz) add a NIC and put it in place of the router.

You don't need to find a specific router, just invoke a firewall, block everything, then allow whatever is requested.

If a request is made and you don't agree with the request, don't allow it.

Job done.

ZoneAlarm will do this:

http://www.zonealarm.com/security/en-us/anti-virus-spyware-free-download.htm

some years back i download a shareware app called "I protect you" that can work as web filter, as well as blocking or allowing only certain sites and or programs to access the internet and could be easily switched on and off via a password. while this works great for one or a few computers, it's a pain to implement to multiple computers as each one has to be configured seperately. i revisted the site after reading your post and found they have a proxy gizmo called ProxEfence that u can install on your server for free that sounds like it may do what your lookin for.
http://www.softforyou.com/prx-index.html