High-order velocity and pressure boundary conditions are presented in Eulerian incompressible smoothed particle hydrodynamics (ISPH). While the high-order convergence of Eulerian ISPH has been demonstrated by the authors for periodic internal flows using Gaussian kernels this was limited by first to second-order accuracy for cases with solid boundaries. Since the SPH interpolation method is numerically robust there is potential for obtaining high-order accuracy in topologically complex domains with robust high-order accurate boundary conditions. In this paper high-order finite-difference extrapolation methods at solid boundaries are developed in Eulerian ISPH to allow for enforcement of the Dirichlet boundary condition for velocity and the Neumann boundary condition for pressure with high-order accuracy. Convergence up to fourth-order is demonstrated for 2-D Taylor-Couette flow and 3-D simulations of Taylor-Couette cellular flow structures are used to demonstrate accuracy and robustness. The order of accuracy may be extended to even higher-order using the analysis presented. Compact fourth-order Wendland-type kernels have also been derived to reduce the particle support region thereby lowering computational effort without loss of high-order convergence. The proposed formulation is therefore entirely high order.