A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. Füredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp Pósa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.