The quantum entangled J/ψ→Σ^{+}Σ[over ¯]^{-} pairs from (1.0087±0.0044)×10^{10} J/ψ events taken by the BESIII detector are used to study the nonleptonic two-body weak decays Σ^{+}→nπ^{+} and Σ[over ¯]^{-}→n[over ¯]π^{-}. The CP-odd weak decay parameters of the decays Σ^{+}→nπ^{+} (α_{+}) and Σ[over ¯]^{-}→n[over ¯]π^{-} (α[over ¯]_{-}) are determined to be 0.0481±0.0031_{stat}±0.0019_{syst} and -0.0565±0.0047_{stat}±0.0022_{syst}, respectively. The decay parameter α[over ¯]_{-} is measured for the first time, and the accuracy of α_{+} is improved by a factor of 4 compared to the previous results. The simultaneously determined decay parameters allow the first precision CP symmetry test for any hyperon decay with a neutron in the final state with the measurement of A_{CP}=(α_{+}+α[over ¯]_{-})/(α_{+}-α[over ¯]_{-})=-0.080±0.052_{stat}±0.028_{syst}. Assuming CP conservation, the average decay parameter is determined as ⟨α_{+}⟩=(α_{+}-α[over ¯]_{-})/2=-0.0506±0.0026_{stat}±0.0019_{syst}, while the ratios α_{+}/α_{0} and α[over ¯]_{-}/α[over ¯]_{0} are -0.0490±0.0032_{stat}±0.0021_{syst} and -0.0571±0.0053_{stat}±0.0032_{syst}, where α_{0} and α[over ¯]_{0} are the decay parameters of the decays Σ^{+}→pπ^{0} and Σ[over ¯]^{-}→p[over ¯]π^{0}, respectively.