The four positive charges at the corners of the square produce a net electric field along the Z-axis directed away from the center of the square (due to symmetry, horizontal components cancel) . A negative charge placed on the Z-axis experiences an attractive force towards the center of the square ($F = qE$) .

For small displacements $z$ (where $z \ll L$), the restoring force is approximately proportional to the displacement ($F \propto -z$), which is the condition for Simple Harmonic Motion (SHM) . Therefore, the negative charge will oscillate back and forth along the Z-axis about the center of the frame.