We investigate the nucleation of superconductivity in a microsquare with a magnetic dot on top. The cusplike behavior of the calculated normal-superconducting phase boundaries, T(c)(H), shows a transition between short-period to long-period oscillations when going from positive to negative applied fields, H. Vorticity changes by more than 1, indicating multiquanta vortex entries, have been detected along this asymmetric T(c)(H) boundary. The dot also expands dramatically the symmetry-consistent vortex-antivortex patterns, thus facilitating their experimental observation.