We analyze $K\to \pi \nu \overline{\nu }$ rates in a model with a TeV-scale leptoquark addressing B-meson anomalies, based on the flavor non-universal 4321 gauge group featuring third-generation quark-lepton unification. We show that, together with the tight bounds imposed by $\Delta F=2$ amplitudes, the present measurement of $B(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ already provides a non-trivial constraint on the model parameter space. In the minimal version of the model, the deviations from the Standard Model in $B(K^{+}\to {\pi }^{+}\nu \overline{\nu })$ are predicted to be in close correlation with non-standard effects in the Lepton Flavor Universality ratios $R_D$ and $R_{D^{⁎}}$. With the help of future data, these correlations can provide a decisive test of the model.