It is well known that \textit{every} Eulerian orientation of an Eulerian $2k$-edge connected (undirected) graph is strongly $k$-edge connected. An important goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity and element connectivity. We show that \textit{every} Eulerian orientation of the hypercube of degree $2k$ is strongly $k$-node connected.