Well, a Cisco Catalyst 3560 could replace our router. The one we have can't!
Am I correct in assuming you're trying to share a single IP with NAT? If that's the case, you require a router, period. Almost all routers come with switches attached to them, but not by definition. A router is simply a device that exchanges data between two (or more) independent networks; usually the dividing line between a LAN and a WAN. You could put to NICs on a linux box and set up routing tables, and it would be a router.
A switch is simply an exchange point. If the switch is managed, it may have an IP address so that it can be configured remotely, but that doesn't give it the ability to do routing.
If you connect a switch directly to the internet, then each of the devices connected to the switch will need their own unique internet IP address. If you take that approach, you don't have a LAN, so it's theoretically possible for (though pretty ridiculous to consider) your local traffic to get routed through the internet, which you definitely don't want.
So, the bottom line is that if you want a cheap solution, you should find a cheap/crappy old computer with a slow processor and little ram, put linux on it, and read up about iptables. You'll have to do a lot of reading/learning if you don't already understand what a router actually does, but it will be worth it in the end.
