The renormalization group equations of the Higgs sector of the minimal nonlinear supersymmetric SU(5) model are derived and numerically solved in the scheme. Evolving the parameters of the model from the GUT scale down to the electroweak scale, the allowed regions of the parameters are determined, in particular the quartic coupling constant λ. The mass bounds, corrections to tree-level mass sum rules and productions of the Higgs bosons at e+e− colliders are investigated for the c.m. energy of LEP1, LEP2 and future linear colliders up to 2000 GeV.