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We use deep narrowband Ca H&K ($F395N$) imaging taken with the Hubble SpaceTelescope (HST) to construct the metallicity distribution function (MDF) ofLocal Group (LG) ultra-faint dwarf (UFD) galaxy Eridanus II (Eri II). Whencombined with archival $F475W$ and $F814W$ data, we measure metallicities for60 resolved red giant branch stars as faint as $m_{F475W}\sim24$ mag, a factorof $\sim4$x more stars than current spectroscopic MDF determinations. We findthat Eri II has a mean metallicity of [Fe/H]$=$-2.50$^{+0.07}_{-0.07}$ and adispersion of $\sigma_{\mbox{[Fe/H]}}=0.42^{+0.06}_{-0.06}$, which areconsistent with spectroscopic MDFs, though more precisely constrained owing toa larger sample. We identify a handful of extremely metal-poor star candidates(EMP; [Fe/H] $< -3$) that are marginally bright enough for spectroscopic followup. Eri II's MDF appears well-described by a leaky box chemical evolutionmodel. We also compute an updated orbital history for Eri II using Gaia eDR3proper motions, and find that it is likely on first infall into the Milky Way.Our findings suggest that Eri II underwent an evolutionary history similar tothat of an isolated galaxy. Compared to MDFs for select cosmologicalsimulations of similar mass galaxies, we find that Eri II has a lower fractionof stars with [Fe/H] $< -3$, though such comparisons should currently betreated with caution due to a paucity of simulations, selection effects, andknown limitations of Ca H&K for EMPs. This study demonstrates the power of deepHST CaHK imaging for measuring the MDFs of UFDs.

Metallicity Distribution Function of the Eridanus~II Ultra-Faint Dwarf Galaxy from Hubble Space Telescope Narrow-band Imaging