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We study the existence and uniqueness problem of compact minimal verticalgraphs in $\mathbb{H}^n\times\mathbb{R}$, $n\geq 2$, over bounded domains inthe slice $\mathbb{H}^n\times\{0\}$, with non-connected boundary having afinite number of $C^0$ hypersufaces homeomorphic to the sphere$\mathbb{S}^{n-1}$, with prescribed bounded continuous boundary data, underhypotheses relating those data and the geometry of the boundary. We show thenonexistence of compact minimal vertical graphs in$\mathbb{H}^n\times\mathbb{R}$ having the boundary in two slices and the heightgreater than or equal to $\pi/(2n-2)$.

Compact minimal vertical graphs with non-connected boundary in $\mathbb{H}^n\times\mathbb{R}$