In my experience the brakes will not work with a broken plastic collar. I'm not sure what to call it either. I've seen quite a few bikes with cracked collars where the brakes simply didn't work because of it. (And when I replaced the collars they worked fine.) When the collar is broken it doesn't "hold in" the spring anymore so whenever the arms are pulled together, instead of the coil spring... coiling... it just bends and pushes out on the broken plastic collar.

Honestly I'd go to your local bike shop or coop and ask if they have any in their spare parts bin. I'm sure they'll have some.

Barring that, V-Brakes (new) are pretty cheap.