We discuss an extragradient projection method for dealing with equilibrium problems involving bifunctions which are strongly quasiconvex on its second argument. The algorithm combines a proximal step with a subgradient projection step using a generalized subdifferential, which is especially useful for dealing with this class of generalized convex functions, and with a line search. As a consequence, the usual assumption regarding the relationship between the Lipschitz-type parameter and the modulus of strong quasiconvexity is no longer needed for ensuring the convergence of the generated sequence to the solution of the problem. Furthermore, numerical experiments for classes of nonconvex mixed variational inequalities based on fractional programming problems are given in order to show the performance of our proposed method.