To understand the origin of the increase in critical current density of rare earth barium cuprate superconductor think films with decreasing thickness, a series of sub-300-nm EuBa2Cu3O7-delta thin films deposited on SrTiO3 substrates are studied by X-ray diffraction and electrical transport measurements. The out-of-plane crystallographic mosaic tilt and the out-of-plane microstrain both increase with decreasing film thickness. The calculated density of c-axis threading dislocations matches the extent of the observed low-field enhancement in critical current density for fields applied parallel to c. The in-plane mosaic twist and in-plane microstrain are both around twice the magnitude of the out-of-plane values, and both increase with decreasing film thickness. The results are consistent with the observed stronger field enhancement in critical current density for fields applied parallel to ab. The lattice parameter variation with thickness is not as expected from consideration of the biaxial strain with the substrate, indicative of in-plane microstrain accommodation by oxygen disorder. Collectively, the results point to an enhancement of critical current by interfacial strain induced oxygen disorder which is greatest closest to the film-substrate interface. The findings of this study have important implications for other thin functional oxide perovskite films and nanostructures where surface and interfacial strains dominate the properties.