In this paper, we consider nonconvex vector equilibrium problems, and discuss the properties of their efficient solution sets. First, based on the Hiriart-Urruty oriented distance function, we introduce a new nonlinear scalarization function, and study its continuity properties. Then, we propose various concepts of connectedness for a vector-valued mapping, and discuss their relationships. Finally, we use these concepts to study sufficient conditions of the nonemptyness and connectedness of weakly and strongly efficient solution sets of such problems via the scalarization method and/or conditions related to the triangle inequality.