ABSTRACT We use deep JWST/NIRSpec R ∼ 1000 slit spectra of 113 galaxies at $1.7 < z < 3.5$, selected from the mass-complete Blue Jay survey, to investigate the prevalence and typical properties of neutral gas outflows at cosmic noon. We detect excess Na id absorption (beyond the stellar contribution) in 46 per cent of massive galaxies (log M*/M⊙ > 10), with similar incidence rates in star-forming and quenching systems. Half of the absorption profiles are blueshifted by at least 100 km s−1, providing unambiguous evidence for neutral gas outflows. Galaxies with strong Na id absorption are distinguished by enhanced emission line ratios consistent with AGN ionization. We conservatively measure mass outflow rates of 3–100 M⊙ yr−1; comparable to or exceeding ionized gas outflow rates measured for galaxies at similar stellar mass and redshift. The outflows from the quenching systems (log(sSFR)[yr−1] ≲ −10) have mass loading factors of 4–360, and the energy and momentum outflow rates exceed the expected injection rates from supernova explosions, suggesting that these galaxies could possibly be caught in a rapid blowout phase powered by the AGN. Our findings suggest that AGN-driven ejection of cold gas may be a dominant mechanism for fast quenching of star formation at z ∼ 2.