Least Squares Finite Element Methods have several appealing properties including a very general formulation, stability, and suitability for multigrid. However, there are disadvantages too, for instance, strong regularity requirements on the exact solution. In particular, the regularity requirements lead to difficulties when solving problems on nonconvex polyhedral domains. In this work we investigate Least Squares Methods based on discontinuous approximation spaces which allow nonconvex polyhedral domains. Applications include first order system formulations of second order elliptic problems and the div-curl problem.