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The quantum entangled J/ψ→Σ+Σ¯− pairs from (1.0087±0.0044)×1010 J/ψ events taken by the BESIII detector are used to study the non-leptonic two-body weak decays Σ+→nπ+ and Σ¯−→n¯π−. The CP-odd weak decay parameters of the decays Σ+→nπ+ (α+) and Σ¯−→n¯π− (α¯−) are determined to be −0.0565±0.0047stat±0.0022syst and 0.0481±0.0031stat±0.0019syst, respectively. The decay parameter α¯− is measured for the first time, and the accuracy of α+ is improved by a factor of four 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 ACP=(α++α¯−)/(α+−α¯−) = −0.080±0.052stat±0.028syst. Assuming CP conservation, the average decay parameter is determined as ⟨α+⟩=(α+−α¯−)/2 = −0.0506±0.0026stat±0.0019syst, while the ratios α+/α0 and α¯−/α¯0 are −0.0490±0.0032stat±0.0021syst and −0.0571±0.0053stat±0.0032syst, where α0 and α¯0 are the decay parameters of the decays Σ+→pπ0 and Σ¯−→p¯π0, respectively.