When a certain material is cooled below its critical temperature,...

For a material exhibiting the Meissner effect (a perfect diamagnet), the magnetic field inside the material is exactly zero ($B = 0$).

The magnetic induction is given by $B = \mu_0 \mu_r H$. Since $B = 0$ and the external field $H \neq 0$, the relative permeability must be zero ($\mu_r = 0$).

Using the relation between relative permeability and magnetic susceptibility, $\mu_r = 1 + \chi$.

Substituting $\mu_r = 0$, we get $0 = 1 + \chi \Rightarrow \chi = -1$.

Thus, the magnetic susceptibility is $-1$ and the relative permeability is $0$.

Answer: $\chi = -1$, $\mu_r = 0$