We explore the possibility that the high energy neutrino flux observed by terrestrial telescopes originates from dark matter (DM) annihilation. Specifically, we study a minimal, UV-complete U (1) extension of the Standard Model with a Dirac DM candidate, whose annihilation into neutrinos proceeds exclusively through a Z ′ boson. By computing the annihilation cross section and comparing with the observed flux, we derive bounds on the model parameters. Additional constraints are obtained within the freeze-in framework, where the observed relic abundance is reproduced. Considering cosmic string vibrations as a source of gravitational waves, we further constrain the vacuum expectation value of the U (1) breaking. All results are contrasted with perturbativity limits and existing constraints from low- and high-energy experiments.