This is about as straightforward as it can get - Newton's Second Law says that the acceleration of an object equals the net force on it divided by its mass:
Newton's Third Law says that isolated forces do not exist - forces always occur in pairs. if the "action force" in this situation is "Hand pushes box.", the reaction force is "Box pushes hand.", right?
So, what is the acceleration of the box? It is still 5 m/s2! Of course! Even though these two forces are equal and opposite, they do not cancel. The force that the hand exerts on the box accelerates the box (at 5 m/s2). The force that the box exerts on the hand accelerates the hand. (Actually, it is the net force on the hand that accelerates the hand, but the box's force contributes to the net force on the hand.)
Forces you exert on other objects accelerate the other objects.