Newton's First Law of Motion
An object with zero net force will have zero acceleration. This does not mean that the object will not be moving, rather its speed would not change. In the case of the box, it is true that, if the person decides to apply 350N on a motionless box that is opposed by 350N of friction, the box would not move. However, since the box is already moving (as implied by the question), a zero net force would mean that the box would not speed up or slow down, i.e. constant velocity.