  We report on the low-temperature thermal expansion and magnetostriction of
the single-impurity quadrupolar Kondo candidate
Y$_{1-x}$Pr$_{x}$Ir$_2$Zn$_{20}$. In the dilute limit, we find a quadrupolar
strain that possesses a singular dependence on temperature $T$,
$\varepsilon_{\mathrm{u}} \sim H^2 \log 1/T$, for a small but finite magnetic
field $H$. Together with the previously reported anomalous specific heat $C$,
this implies a quadrupolar Gr\"uneisen ratio $\Gamma_u = \partial_T
\varepsilon_{\mathrm{u}}/C \sim H^2/(T^2 \log 1/T)$ whose divergence for finite
$H$ is consistent with the scenario of a quadrupolar Kondo effect. In addition,
we find a singular behavior of the isotropic strain $\varepsilon_{\mathrm{B}}$
in zero magnetic field resulting in a divergence of both the volume thermal
expansion and the volume Gr\"uneisen parameter. We speculate that this behavior
might be also induced by putative Kondo correlations via elastic
anharmonicities or static strain disorder.
