Several dozen hypervelocity star (HVS) candidates have been reported based on the second data release of Gaia (Gaia DR2). However, it has been proven that the radial velocities of some Gaia HVS candidates are not reliable. In this paper, we employ refined astrometric criteria to re-examine Gaia DR2, arriving at a more reliable sample of HVS and high velocity star candidates than those found by previous authors. We develop a method called Binary Escape Probability Analysis to identify some HVS candidates. This method allows us to work with stars having only two epochs of measured radial velocity. These stars were usually discarded in previous similar studies. A scrutiny of our final results sheds light on selection effects present in our studies, which we propose to be the focus of future studies. In total, we find three late-type (2 G-type and 1 K-type) HVS and 21 high velocity star candidates, 3 and 11 of which are new, respectively. Judging by their historical trajectories, which we calculate, all three HVS candidates could not have had Galactic center origins. Further monitoring is required to confirm their status.