We consider a class of Hamilton-Jacobi equations H(x, Du(x)) - 0 with no u-dependence and with continuity properties consistent with recent applications in queueing theory. Continuous viscosity solutions are considered in a compact polyhedral domain, with oblique derivative (Neumann-type) boundary conditions. Comparison and uniqueness results are presented, which use monotonicity of H(x, p) in the p variable for values of p in the appropriate sub-and superdifferential sets of the solution u(x). Several examples illustrate the results.