The stability argument is as follows--- the Geon system will have some mass, and it is made out of massless fields orbiting in closed orbits, so if you make the geon a little smaller with the same total energy, you expect the gravity to win and the massless fields to collapse into a black hole, and if you make the geon a little bigger, you expect the massless stuff to disperse to infinity.
This argument is hard to make rigorous, because you need to find a way to rescale the nonlinear gravitational theory. So Wheeler studied this situation extensively, with the hope of finding a stable Geon. He didn't find one, and even if there were one, we already have a good model of elementary particles in the black hole solutions and their quantum counterparts, so it is not clear that such a solution would be useful.
But it is a strangely neglected field. Perhaps there is an easy argument that establishes instability of all geons, but it is going to be tough, because the Geons can make arbitrarily complicated links of light going through each other, pulling each other into stable orbits.