We investigate the evolution of supernova remnants (SNRs) in the early universe by solving the one-dimensional hydrodynamics and non-equilibrium chemistry under spherical symmetry. A cooled dense shell, which is formed behind the supernova blastwave during snowplow phase, is thought to be an important site where star formation activity occurs after shell fragmentation. To evaluate the characteristic fragmentation mass of the shell, we calculate the thermal and chemical evolution of the shell. We show that the gas within the shell cools by H2 or HD line cooling in the metal-free or extremely low-metallicity environments (≤10-4Z⊙(, while mainly by the fine-structure line cooling of CI and OI in the more metal-enriched environment (≥10-3Z⊙). We then study the shell fragmentation using the linear stability analysis of an expanding shell. Since the gas mass which can cool by HD or metal line cooling is very low, shell fragmentation is unlikely to occur unless the ambient temperature is very low (100 K). The shell fragmentation thus strongly depends on the densities and temperatures of the ambient medium and the supernova energy but depends weakly on the metallicity.