We report on a search for the standard model Higgs boson produced in association with a $W$ or $Z$ boson in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ recorded by the CDF II experiment at the Tevatron in a data sample corresponding to an integrated luminosity of $2.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. We consider events which have no identified charged leptons, an imbalance in transverse momentum, and two or three jets where at least one jet is consistent with originating from the decay of a $b$ hadron. We find good agreement between data and background predictions. We place 95% confidence level upper limits on the production cross section for several Higgs boson masses ranging from $110\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ to $150\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. For a mass of $115\text{ }\text{ }\mathrm{GeV}/{c}^{2}$ the observed (expected) limit is 6.9 (5.6) times the standard model prediction.