A generalization of the box-variational method is developed to describe particle scattering with the Dirac equation. The method is applied to extract phase shifts from all-order single + double relativistic many-body perturbation theory calculations of the electron-helium, electron-neon, and electron-krypton systems. Comparisons with experimental elastic and momentum transfer cross sections are made. Agreement at the 1% to 2% level is achieved for helium and neon. The scattering length for krypton is 4% smaller in magnitude than experimental estimates derived from swarm experiments.