You can pull a wagon with a rope, but you can't push it with a rope. Is there such a thing as a "negative" tension?

When you pull a wagon with a rope, the rope tightens up and transfers your force on to the wagon. If the magnitude of the force is enough to displace the wagon, it will start to move. However when you pushes on the rope, it bends, and the force you apply does not acts on the wagon, it is as if the wagon has not experienced any force.
Therefore the tension in the string can be interpreted as a measure of the pulling force exerted by a string on to an object, and in that context the tension in the string can never be negative. It can either be zero or it can be positive.