In this talk, I will consider the approximation of eigenvalues for Maxwell's systems based on a very weak formulation of the problem. This involves two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is (L²(D))³ and the data resides in various negative norm spaces as done in [BP]. A finite element least-squares approximation to these variational problems will be used as a basis for the eigenvalue approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues will be set up involving only the approximation to either of the div-curl systems. We will give some theorems which guarantee the convergence of the eigenvalues to those of the continuous problem without the occurence of spirous values. All of the results are valid for problems with low Sobolev regularity, e.g., when the eigenfunctions are only in (H^s(D))⁶, for s only slightly greater than zero. Such low regularity solutions are possible in applications such as those involving anteannas in R³. Finally, some results of numerical experiments will be reported.