I'm going to tentatively say, "no, horror doesn't need an unhappy ending," but I'm as tentative about that as a man contemplating going down into the cellar of the house he just bought, even though he doesn't have a flashlight and he might have a dark secret in his past. Or maybe he flatlines for fun with his friends and he's worried he brought something back?
All horror really needs is some sort of anticipation-release mechanism for the viewer/reader. Which shouldn't surprise us as that's the same required mechanism of melodrama and pornography--the other two "body genres" that Linda Williams identifies. Only in horror, the anticipation-release has to do with fear.
And yet, let's note that the anticipation-release in those genres comes at the end: That is, you may cry during melodrama, but you're meant to leave the theater crying; similarly with pornography, when you're done, you're done. So, transposing this to horror, we could say that the big release should come towards the end. Which is another way of saying there shouldn't be a lot of denouement between the climax and the credits.
But when a work of horror ends on a hopeful note, the big release is a different flavor than if we end on a sad or on a mixed note. That's pretty close to a tautology: a happy ending makes you feel different than a sad ending. But I think it's worth noting that our genre notion of horror can take in these different flavors: the protagonist fails (morally, physically), the protagonist succeeds in sacrificing, the protagonist succeeds.