A least squares mixed finite element method for deformations of hyperelastic materials using stress and displacement as process variables is presented and studied. The method is investigated in detail for the specific case of a neo-Hookean material law and is based on the representation of the strain-stress relation. A formulation is derived for compressible materials and shown to remain valid in the incompressible limit, automatically enforcing the incompressibility constraint. The mapping properties of the first-order system operator are studied in appropriate Sobolev spaces. Under the assumption of a locally unique solution with sufficient regularity, it is proved that the first-order least squares residual constitutes an upper bound for the error measured in a suitable norm, provided that the finite element approximation is sufficiently close. The method is tested numerically in a plane strain situation using next-to-lowest-order Raviart--Thomas elements for the stress tensor and conforming quadratic ...