The Hitch-Hiker(1953)

Roy and Gilbert's fishing trip takes a terrifying turn when the hitchhiker they pick up turns out to be a sociopath on the run from the law. He's killed before, and he lets the two know that as soon as they're no longer useful, he'll kill again. The two friends plot an escape, but the hitchhiker's peculiar physical affliction, an eye that never closes even when he sleeps, makes it impossible for them to tell when they can make a break for it.