We present the mass excesses of Sc52-57, obtained from recent time-of-flight nuclear mass measurements at the National Superconducting Cyclotron Laboratory at Michigan State University. The masses of Sc56 and Sc57 were determined for the first time with atomic mass excesses of -24.85(59)(-54+0)MeV and -21.0(1.3)MeV, respectively, where the asymmetric uncertainty for Sc56 was included due to possible contamination from a long-lived isomer. The Sc56 mass indicates a small odd-even mass staggering in the A=56 mass chain towards the neutron drip line, significantly deviating from trends predicted by the global FRDM mass model and favoring trends predicted by the UNEDF0 and UNEDF1 density functional calculations. Together with new shell-model calculations of the electron-capture strength function of Sc56, our results strongly reduce uncertainties in model calculations of the heating and cooling at the Ti56 electron-capture layer in the outer crust of accreting neutron stars. We find that, in contrast to previous studies, neither strong neutrino cooling nor strong heating occurs in this layer. We conclude that Urca cooling in the outer crusts of accreting neutron stars that exhibit superbursts or high temperature steady-state burning, which are predicted to be rich in A≈56 nuclei, is considerably weaker than predicted. Urca cooling must instead be dominated by electron capture on the small amounts of adjacent odd-A nuclei contained in the superburst and high temperature steady-state burning ashes. This may explain the absence of strong crust Urca cooling inferred from the observed cooling light curve of the transiently accreting x-ray source MAXI J0556-332.