We present 20 candidate hypervelocity stars from the Sloan Extension for Galactic Understanding and Exploration (SEGUE) G and K dwarf samples. Previous searches for hypervelocity stars have only focused on large radial velocities; in this study, we also use proper motions to select the candidates. We determine the hypervelocity likelihood of each candidate by means of Monte Carlo simulations, considering the significant errors often associated with high proper motion stars. We find that nearly half of the candidates exceed their escape velocities with at least 98% probability. Every candidate also has less than a 25% chance of being a high-velocity fluke within the SEGUE sample. Based on orbits calculated using the observed six-dimensional positions and velocities, few, if any, of these candidates originate from the Galactic center. If these candidates are truly hypervelocity stars, they were not ejected by interactions with the Milky Way's supermassive black hole. This calls for a more serious examination of alternative hypervelocity-star ejection scenarios.