If your current ethernet cables are dismantle-able then yes, you can use the connectors from the ethernet cable on the phone cable

but if not, you can buy RJ45 connectors for a few pence at your local computer bits shop (radioshack? I think that's the standard US computer bits shop?)

You could use the existing phone connectors and connect the machines over modems
but you'll have to setup dialup networking and it'll be dog slow

for the 10 mins it'll take to chop the connectors off, strip the wires back, crimp the wires to RJ45 (ethernet) connectors and snap the RJ45 connectors shut, it's worth making a home-made CAT3 cable