where $\mu_0 H$ is the applied field, $\mu_0H_c$ the critical field and $M$ the magnetization. In general they are related by $$\vec B=\mu_0(\vec H+\vec M).$$ Bellow the critical field the measured field in the sample vanishes ($B=0$), and therefore $M=-H$.
Above the critical field the material should behave as a normal conductor. Then why does $M$ vanish above the critical field?
I would rather assume the material is either diamagnetic, paramagnetic or ferromagnetic in this limit. In any case, the presence of an applied field would give raise to a non zero magnetization. For diamagnetic or paramagnetic materials one has $\vec M=\chi_m \vec H$ and since $\chi_m$ is too small then $M\approx 0$. But $\vec M$ is by no means negligible for a ferromagnetic material.