This paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi-Douglas-Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart-Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuska-Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.