Feedback in Emerging Extragalactic Star Clusters (JWST--FEAST): Calibration of Star Formation Rates in the Mid-Infrared with NGC 628

New JWST near-infrared imaging of the nearby galaxy NGC 628 from the Cycle 1 program JWST-FEAST is combined with archival JWST mid-infrared imaging to calibrate the 21 $\mu$m emission as a star formation rate indicator (SFR) at $\sim$120 pc scales. The Pa$\alpha$ ($\lambda$1.8756 $\mu$m) hydrogen recombination emission line targeted by FEAST provides a reference SFR indicator that is relatively insensitive to dust attenuation, as demonstrated by combining this tracer with the HST H$\alpha$ imaging. Our analysis is restricted to regions that appear compact in nebular line emission and are sufficiently bright to mitigate effects of both age and stochastic sampling of the stellar initial mass function. We find that the 21 $\mu$m emission closely correlates with the nebular line emission, with a power-law with exponent=1.07$\pm$0.01, in agreement with past results. We calibrate a hybrid SFR indicator using a combination of H$\alpha$ and 24 $\mu$m (extrapolated from 21 $\mu$m) tracers and derive the proportionality constant between the two tracers $b=0.095\pm0.007$, which is $\sim$ 3-5 times larger than previous derivations using large regions/entire galaxies. We model these discrepancies as an increasing contribution to the dust heating by progressively older stellar populations for increasing spatial scales, in agreement with earlier findings that star formation is hierarchically distributed in galaxies. Thus, use of hybrid SFR indicators requires prior knowledge of the mean age of the stellar populations dominating the dust heating, which makes their application uncertain. Conversely, non-linear calibrations of SFRs from L(24) alone are more robust, with a factor $\lesssim$2.5 variation across the entire range of L(24) luminosities from HII regions to galaxies.