We present the near-infrared spectra (2.5-5.0 um) of shocked $H_2$ gas, observed with the Infrared Camera onboard the satellite AKARI. Two supernova remnants, IC 443 and HB 21, were observed. IC 443 shows a hint of non-equilibrium ortho-to-para ratio (OPR): 2.4 (-0.2, +0.3). HB 21 also shows an indication of a potential non-equilibrium OPR: 1.8-2.0. These non-equilibrium OPRs are first reported for shocked $H_2$ gas at E(v,J) > 7000 K, as far as we are aware. We concluded that the non-equilibrium OPR probably originates from dissociative J-shocks, considering several factors such as the shock combination requirement, the line ratios, and the possibility that $H_2$ gas can form on grains with a non-equilibrium OPR. The difference in the collision energy of H atoms on grain surfaces would give rise to the observed difference between the OPRs of IC 443 and HB 21, if dissociative J-shocks are responsible for the $H_2$ emission. Our study suggests that shocked-then-cooled $H_2$ gas may play as a heat reservoir with the non-equilibrium OPR.