  The quantum entangled $J/\psi \to \Sigma^{+}\bar{\Sigma}^{-}$ pairs from
$(1.0087\pm0.0044)\times10^{10}$ $J/\psi$ events taken by the BESIII detector
are used to study the non-leptonic two-body weak decays $\Sigma^{+} \to n
\pi^{+}$ and $\bar{\Sigma}^{-} \to \bar{n} \pi^{-}$. The $C\!P$-odd weak decay
parameters of the decays $\Sigma^{+} \to n \pi^{+}$ ($\alpha_{+}$) and
$\bar{\Sigma}^{-} \to \bar{n} \pi^{-}$ ($\bar{\alpha}_{-}$) are determined to
be $-0.0565\pm0.0047_{\rm stat}\pm0.0022_{\rm syst}$ and $0.0481\pm0.0031_{\rm
  stat}\pm0.0019_{\rm syst}$, respectively. The decay parameter
$\bar{\alpha}_{-}$ is measured for the first time, and the accuracy of
$\alpha_{+}$ is improved by a factor of four compared to the previous results.
The simultaneously determined decay parameters allow the first precision $C\!P$
symmetry test for any hyperon decay with a neutron in the final state with the
measurement of
$A_{C\!P}=(\alpha_{+}+\bar{\alpha}_{-})/(\alpha_{+}-\bar{\alpha}_{-})=-0.080\pm0.052_{\rm
  stat}\pm0.028_{\rm syst}$. Assuming $C\!P$ conservation, the average decay
parameter is determined as $\left< \alpha_{+}\right>=(\alpha_{+}-
\bar{\alpha}_{-})/2 = -0.0506\pm0.0026_{\rm stat}\pm0.0019_{\rm syst}$, while
the ratios $\alpha_{+}/\alpha_{0}$ and $\bar{\alpha}_{-}/\bar\alpha_{0}$ are
$-0.0490\pm0.0032_{\rm stat}\pm0.0021_{\rm syst}$ and $-0.0571\pm0.0053_{\rm
stat}\pm0.0032_{\rm syst}$, where $\alpha_{0}$ and $\bar\alpha_{0}$ are the
decay parameters of the decays $\Sigma^{+} \to p \pi^{0}$ and $\bar{\Sigma}^{-}
\to \bar{p} \pi^{0}$, respectively.
