We carried out a spectroscopic observation of the BL Lacertae object Mrk 421 with the Extreme Ultraviolet Explorer (EUVE) over an 11 day period in 1995 late April to early May (~242 ks useful time). During this period, the source underwent a flare that was detected also in X-rays and TeV γ-rays. The best continuous coverage of the flare was obtained by EUVE, which resolved the smooth rise and fall of the flux, measuring a variability of as much as a factor of ~1.5 over a span of ~2 days. The detected spectrum extended from ~65 to 100 Å and could be fitted with a power law of energy spectral index αEUV ~ 3.5 +/- 0.8 for the measured Galactic hydrogen column density. The EUV spectrum is much steeper than the mean 1.5-7.5 keV X-ray spectrum, αX = 1.63 +/- 0.02, measured simultaneously by the Advanced Satellite for Cosmology and Astrophysics. Furthermore, a simple power-law fit to the observed fluxes at 85 Å and 1.5 keV (excluding the data from the first 4 days, the time of maximum variability) significantly overestimates the flux at the shortest detected EUV wavelengths. These two findings imply that strong absorption is occurring between ~65 and ~75 Å. Such absorption is quite similar to that detected previously in our observation of the BL Lacertae object PKS 2155-304. We demonstrate that this absorption can be attributed to a superposition of Doppler-smeared absorption lines originating in high-velocity, QSO-type nuclear clouds of total column density ~5 × 1021 cm-2 that are ionized by the beamed continuum of the associated relativistic jet. We identify the lines as mostly L- and M-shell transitions of Mg and Ne. The data suggest that the velocity range spanned by the clouds is relatively small (from vi ~ 0.05c to vf ~ 0.1c). We find that such a range is consistent with a scenario in which the clouds are initially accelerated to vi by a magnetized outflow from a nuclear accretion disk, with radiation pressure further accelerating them to vf after they enter the beamed emission cone of the jet. We also compute the expected cloud absorption lines in the UV and soft X-ray regimes and use these results to constrain the physical parameters of the clouds and their chemical composition.