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, and they all showed the ortho-to-para ratios (OPRs) of less than 3.0: 2.1-2.2 for IC 443 and 1.6-1.8 for HB 21. These non-equilibrium OPRs are first reported at E(v,J) > 7000 K, as far as we are aware of. Based on our previous study, we try to interpret that the non-equilibrium OPRs originate from dissociative J-shocks. Dissociative J-shocks mainly generate infrared H2 emissions from their $H_2$ reformation zone, and the OPR of 3.0 are expected for the reformed $H_2$ from the theoretical study. This is contradictory to our observational results. We propose other possible origins of the non-equilibrium OPRs, such as, abnormal $H_2$ reformation, partially dissociative J-shocks, etc.