A search is presented for single production of heavy vector-like quarks (B) that decay to a Higgs boson and a b quark, with the Higgs boson decaying to a highly boosted $\mathrm{b\overline{b}}$ pair reconstructed as a single collimated jet. The analysis is based on data collected by the CMS experiment in proton-proton collisions at $\sqrt{s} =$ 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The data are consistent with background expectations, and upper limits at 95% confidence level on the product of the B quark cross section and the branching fraction are obtained in the range 1.28-0.07 pb, for a narrow B quark with a mass between 700 and 1800 GeV. The production of B quarks with widths of 10, 20 and 30% of the resonance mass is also considered, and the sensitivities obtained are similar to those achieved in the narrow width case. This is the first search at the CERN LHC for the single production of a B quark through its fully hadronic decay channel, and the first study considering finite resonance widths of the B quark.