Yes, I expect you can - plus you may be able to use some of the other features. However, if you've got a Linksys router, you won't need to use a machine as a router, as that is obviously ideal for the job. (Linksys routers are little Linux appliances,I think, like most of those routers).If you don't have one already, than a plain switch should be quite a bit cheaper especially online.
At home, for example I've got a cable internet modem, which is connected to a similar router to yours. Hanging off the router are 2 machines, a Toshiba laptop & an old desktop that is an NFS server. NAT & DHCP is enabled on the router, so the router hands out '192.168....' type addresses & the Network Address Translation routes the internet stuff to the correct machine, while they aren't visible from the internet. Those cheap home routers are handy for building a home network as routing & DHCP is a lot easier to set up on the routers web-based GUI than doing it from scratch on a machine,plus you gain another machine. Mind you, if you want to do it all yourself for the fun of it, that's another matter