  A supramassive, strongly-magnetized millisecond neutron star (NS) has been
proposed to be the candidate central engine of at least some short gamma-ray
bursts (SGRBs), based on the "internal plateau" commonly observed in the early
X-ray afterglow. While a previous analysis shows a qualitative consistency
between this suggestion and the Swift SGRB data, the distribution of observed
break time $t_b$ is much narrower than the distribution of the collapse time of
supramassive NSs for the several NS equations-of-state (EoSs) investigated. In
this paper, we study four recently-constructed "unified" NS EoSs, as well as
three developed strange quark star (QS) EoSs within the new confinement
density-dependent mass model. All the EoSs chosen here satisfy the recent
observational constraints of the two massive pulsars whose masses are precisely
measured. We construct sequences of rigidly rotating NS/QS configurations with
increasing spinning frequency $f$, from non-rotating ($f = 0$) to the Keplerian
frequency ($f = f_{\rm K}$), and provide convenient analytical parametrizations
of the results. Assuming that the cosmological NS-NS merger systems have the
same mass distribution as the Galactic NS-NS systems, we demonstrate that all
except the BCPM NS EoS can reproduce the current $22\%$ supramassive NS/QS
fraction constraint as derived from the SGRB data. We simultaneously simulate
the observed quantities (the break time $t_b$, the break time luminosity $L_b$
and the total energy in the electromagnetic channel $E_{\rm total}$) of SGRBs,
and find that while equally well reproducing other observational constraints,
QS EoSs predict a much narrower $t_b$ distribution than that of the NS EoSs,
better matching the data. We therefore suggest that the post-merger product of
NS-NS mergers might be fast-rotating supramassive QSs rather than NSs.
