We have studied the microstructure of latex suspensions formulated with hydrophobically modified ethylene (oxide) urethane (HEUR) thickener (or rheology modifier, RM) using small-angle neutron scattering under shear (rheo-SANS). Within the shear rate range studied (0-1000 s(-1)), the neutron scattering profiles are consistent with a polydisperse core-shell model, with the latex particles comprising the core and an adsorbed layer of water-swollen RM on the latex surface forming the shell. The core-shell structure is isotropic under quiescent conditions but becomes anisotropic under shear (with the major axis along the vorticity direction). During shear, the solvent (D2O/H2O) is expelled (hydrodynamic squeezing) from the swollen polymer chains, and the shell structure becomes denser. The anisotropic shell is a result of differing degrees of compression along the flow and vorticity directions. With increasing shear rate, the shell thickness (in both the flow and vorticity direction) tends toward asymptotic values (with the shell thickness in the vorticity direction greater than the shell thickness in the flow direction) independent of the RM hydrophobe density (defined as the average number of hydrophobes per polymer chain). The RM concentration (w/w) in the adsorbed layer varies from similar to 0.05-0.1(at low shear) to similar to 0.25-0.4 (high shear, similar to 1000 s(-1)) with higher values for the RM polymer with higher hydrophobe density. The swollen RM-water shell substantially increases the effective volume fraction of the dispersed latex particles. We find, however, that accounting for this increase within the conventional effective hard-sphere (Krieger-Dougherty) dispersion rheology model does not fully explain the higher viscosity of the formulated mixture. We hypothesize the existence of latex-latex interactions mediated by RM polymer bridges even at high shear. The large-scale structure of the particle assembly will be reported in a subsequent manuscript.