The Jones eigenvalue problem is an overdetermined problem, where the Neumann eigenvalue problem for linear elasticity is coupled with a constraint on the normal trace of the displacement along the boundary. This eigenvalue problem presents interesting features, not least of which is the sensitive dependance on boundary geometry. We prove the existence of eigenpairs of this eigenvalue problem on Lipschitz domains in 2D and 3D, and use numerical methods to approximate the eigenpairs on some simple geometries.