Magnetic spin-down of a millisecond neutron star has been proposed as the power source of hydrogen-poor superluminous supernovae (SLSNe-I). However, producing an unambiguous test that can distinguish this model from alternatives, such as circumstellar interaction, has proven challenging. After the supernova explosion, the pulsar wind inflates a hot cavity behind the expanding stellar ejecta: the nascent millisecond pulsar wind nebula. Electron/positron pairs injected by the wind cool through inverse Compton scattering and synchrotron emission, producing a pair cascade and hard X-ray spectrum inside the nebula. These X-rays ionize the inner exposed side of the ejecta, driving an ionization front that propagates outwards with time. Under some conditions this front can breach the ejecta surface within months after the optical supernova peak, allowing ~0.1-1 keV photons to escape the nebula unattenuated with a characteristic luminosity L_X ~ 1e43-1e45 erg/s. This ionization break-out may explain the luminous X-ray emission observed from the transient SCP 06F, providing direct evidence that this SLSN was indeed engine-powered. Luminous break-out requires a low ejecta mass and that the spin-down time of the pulsar be comparable to the photon diffusion timescale at optical maximum, the latter condition similar to that required for a supernova with a high optical fluence. These relatively special requirements may explain why most SLSNe-I are not accompanied by detectable X-ray emission. Global asymmetry of the supernova ejecta increases the likelihood of an early break-out along the direction of lowest density. Atomic states with lower threshold energies are more readily ionized at earlier times near optical maximum, allowing UV break-out across a wider range of pulsar and ejecta properties than X-ray break-out, possibly contributing to the blue/UV colors of SLSNe-I.