The effect of edges and apertures on the Casimir energy of an arrangement of plates and boundaries can be calculated in terms of an effective nonlocal lower-dimensional field theory that lives on the boundary. This formalism has been developed in a number of previous papers and applied to specific examples with Dirichlet boundary conditions. Here we generalize the formalism to arbitrary boundary conditions. As a specific example, the geometry of a flat plate and a half-plate placed parallel to it is considered for a number of different boundary conditions and the area-dependent and edge dependent contributions to the Casimir energy are evaluated. While our results agree with known results for those special cases (such as the Dirichlet and Neumann limits) for which other methods of calculation have been used, our formalism is suitable for general boundary conditions, especially for the diffractive effects.