In the conventional description of strong-field high-harmonic generation (HHG), the atom is treated quantum mechanically while the driving laser is represented by a classical field. Quantum fluctuations of the driver can become relevant in nanoscale interaction geometries, where tight spatial confinement can raise the vacuum-field amplitude to $\sim {10}^{-2}$ of the mean driving-field amplitude. In this regime, fluctuations of a squeezed driver can imprint its quadrature statistics on HHG observables. To model this interaction, we sample the positive Wigner function of the driving field and solve the time-dependent Schr"odinger equation for each sampled field realization. Within a single-mode Gaussian framework, we show that amplitude squeezing suppresses the shot-to-shot variance of the HHG cutoff below the coherent-state standard quantum limit (SQL) at the same mean field. To leading order in the vacuum-to-driver field ratio, the normalized variance follows $e^{-2r}$, where $r$ is the squeezing parameter, independently of the absolute vacuum-field amplitude. At stronger squeezing, quadratic coupling of the anti-squeezed quadrature to the field modulus competes with this suppression, yielding a variance minimum near $r_{opt}\approx 1.55$ for the parameters studied. A sub-SQL cutoff variance thus indicates sub-vacuum noise in the quadrature probed by the cutoff, suggesting that shot-resolved HHG cutoff measurements could witness driving-field nonclassicality.