In this paper, we construct generic, spherically symmetric thin-shell wormholes and check their stabilities using the unified dark sector, including dark energy and dark matter. We give a master equation, from which one can recover, as a special case, other stability solutions for generic spherically symmetric thin-shell wormholes. In this context, we consider a particular solution; namely we construct an effective thin-shell wormhole under Lorentz symmetry breaking. We explore stability analyses using different models of the modified Chaplygin gas with constraints from cosmological observations, such as seventh-year full Wilkinson microwave anisotropy probe data points, type Ia supernovae, and baryon acoustic oscillation. In all these models we find stable solutions by choosing suitable values for the parameters of the Lorentz symmetry breaking effect.