Patchy nanocubes are intriguing materials with simple shapes and space-filling and multidirectional bonding properties. Previous studies have revealed various mesoscopic structures such as colloidal crystals in the solid regime and rod-like or fractal-like aggregates in the liquid regime of the phase diagram. Recent studies have also shown that mesoscopic structural properties, such as an average cluster size M and orientational order, in amphiphilic nanocube suspensions are associated with macroscopic viscosity changes, mainly owing to differences in cluster shape among patch arrangements. Although many studies have been conducted on the self-assembled structures of nanocubes in bulk, little is known about their self-assembly in nanoscale spaces or structural changes under shear. In this study, we investigated mixtures of one- and two-patch amphiphilic nanocubes confined in two flat parallel plates at rest and under shear using molecular dynamics simulations coupled with multiparticle collision dynamics. We considered two different patch arrangements for the two-patch particles and two different slit widths H to determine the degree of confinement in constant volume fractions in the liquid regime of the phase diagram. We revealed two unique cluster morphologies that have not been previously observed under bulk conditions. At rest, the size of the rod-like aggregates increased with decreasing H, whereas that of the fractal-like aggregates remained constant. Under weak shear with strong confinement, the rod-like aggregates maintained a larger M than the fractal-like aggregates, which were more rigid and maintained a larger M than the rod-like aggregates under bulk conditions.