We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) $Δ$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $Δ$-regular tree. We also find efficient counting and sampling algorithms for proper $q$-colorings of random $Δ$-regular bipartite graphs when $q$ is sufficiently small as a function of $Δ$.