Using neutron reflectivity and the newly developed Shear Confinement Cell (SCC), we have directly quantified the density distribution of opposing neutral polymer brushes confined between parallel plates in good solvent conditions. With an average separation between the plates of approximately 1000 Å, our measurements show that the density profile in the overlap region between opposing polymer brushes flattens consistent with predictions from molecular dynamics simulations. A significant increase in density at the anchoring surfaces due to compression of the brush layers is observed. This compression or collapse of the brushes in restricted geometries strongly suggests that high-density brushes do not interpenetrate significantly in good solvent conditions. In addition, for the first time, we have measured the effects of an applied shear stress on the sample. We find that for neutral brushes, shear creates a totally new disentangled structure which surprisingly relaxes only after a time span of a few weeks.