Yes, a cat could kill a full sized hen....not sure about a rooster.
I would surmise that a nice big, flock protecting, rooster would probably intimidate a cat.
You'll want to maintain good relations with the neighbor's. So have a nice friendly talk with the neighbors about their cats and their relationship with your chickens. That should take care of it.
If that fails, then there are a whole host of options, some good....some not do good.