Recent inelastic neutron scattering (INS) measurements on FeSe and Fe(Te1−xSex) have sparked intense debate over the nature of the ground state in these materials. Here we propose an effective bilinear-biquadratic spin model, which is shown to consistently describe the evolution of low-energy spin excitations in FeSe, both under applied pressure and upon Se/Te substitution. The phase diagram, studied using a combination of variational mean-field, flavor-wave calculations and density-matrix renormalization group (DMRG), exhibits a sequence of transitions between the columnar antiferromagnet common to the iron pnictides, the nonmagnetic ferroquadrupolar phase attributed to FeSe, and the double-stripe antiferromagnetic order known to exist in Fe1+yTe. The calculated spin structure factor in these phases mimics closely that observed with INS in the Fe(Te1−xSex) series. In addition to the experimentally established phases, the possibility of incommensurate magnetic order is also predicted.