In a square loop current, the vector potential can be zero at specific points due to the symmetry of the loop. The discussion clarifies that the vector potential is indeed zero at the center of the loop, not just along the two sides parallel to the x-axis. This phenomenon occurs because the contributions from opposite sides of the loop cancel each other out. The focus is on understanding how the geometry of the loop influences the vector potential. Overall, the cancellation effects of the current in the square loop lead to a zero vector potential at the center.