We report on the results of an experimental investigation into the newly developed Yb0.19Y0.34Lu0.12Gd0.35Ca4O(BO3)3 mixed oxyborate crystal as a promising Yb-ion laser material for applications in microchip lasers. Highly efficient continuous-wave (CW) laser action was demonstrated with 0.60 mm thick crystal plates in a microchip resonator under 976-nm diode pumping. With 4.80 W of pump power absorbed, an output power of 3.15 W was produced at wavelengths around 1057 nm, resulting in an optical-to-optical efficiency of 65.5%; while in the low pumping region this efficiency could be as high as 83.4%. A comparative study confirms the superiority of this mixed oxyborate to the parent compound of Yb:YCOB in CW microchip laser performance. Stable passively Q-switched laser operation was achieved with a 2D WS2 saturable absorber incorporated into the microchip resonator, generating a pulsed output power of 1.20 W at a repetition rate of 404 kHz; the resulting pulse energy, duration, and peak power were, respectively, 3.0 μJ, 86 ns, and 34.9 W. The results presented show the great potential of this Yb-doped mixed oxyborate crystal, in the development of CW or passively Q-switched microchip lasers operating in the 1-μm near-infrared region.