  We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times
\mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some
uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or
asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two
ended complete hypersurfaces.
