A search is presented for decays of (Z) and Higgs bosons to a (J/\psi ) meson and a photon, with the subsequent decay of the (J/\psi ) to ({\mu }^{+}{\mu }^{-}). The analysis uses data from proton-proton collisions with an integrated luminosity of 35.9 (\phantom{\rule{0.166667em}{0ex}}{\text{fb}}^{-1}) at (\sqrt{s}=13\phantom{\rule{0.166667em}{0ex}}\text{TeV}) collected with the CMS detector at the LHC. The observed limit on the (Z\to J/\psi \gamma ) decay branching fraction, assuming that the (J/\psi ) meson is produced unpolarized, is (1.4×{10}^{-6}) at 95% confidence level, which corresponds to a rate higher than expected in the standard model by a factor of 15. For extreme-polarization scenarios, the observed limit changes from (-13.6) to (+8.6%) with respect to the unpolarized scenario. The observed upper limit on the branching fraction for (H\to J/\psi \gamma ) where the (J/\psi ) meson is assumed to be transversely polarized is (7.6×{10}^{-4}), a factor of 260 larger than the standard model prediction. The results for the Higgs boson are combined with previous data from proton-proton collisions at (\sqrt{s}=8\phantom{\rule{0.166667em}{0ex}}\text{TeV}) to produce an observed upper limit on the branching fraction for (H\to J/\psi \gamma ) that is a factor of 220 larger than the standard model value.