The Higgs boson spectrum of the next-to-minimal supersymmetric standard model is examined. The model includes a singlet Higgs field S in addition to the two Higgs doublets of the minimal extension. 'Natural' values of the parameters of the model are motivated by their renormalization group running and the vacuum stability. The qualitative features of the Higgs boson masses are dependent on how strongly the Peccei-Quinn U(1) symmetry of the model is broken, measured by the self-coupling of the singlet field in the superpotential. We explore the Higgs boson masses and their couplings to gauge bosons for various representative scenarios.