What do you call a person who does not agree with anything?

By definition, someone who does not agree with anything would be disagreeable. Though argumentative may also describe such a person, depending on the situation.

There is a southern tradition of this. Some people just will argue any point, and in southern culture we refer to them as a "Contrary." a Contrary will argue his own name with you, will argue that water ain't wet and sugar ain't sweet.

There's no rhyme nor reason to it; they're simply born that way. Any discussion with a Contrary will eventually turn into a debate.