Observations of high-redshift galaxies (z>5) have shown that these galaxies have extreme emission lines with equivalent widths much larger than their local star-forming counterparts. Extreme emission line galaxies (EELGs) in the nearby universe are likely analogs to galaxies during the Epoch of Reionization and provide nearby laboratories to understand the physical processes important to the early universe. We use Hubble Space Telescope/Cosmic Origins Spectrograph and Large Binocular Telescope/Multi-Object Double Spectrographs spectra to study two nearby EELGs, J104457 and J141851. The far-UV spectra indicate that these two galaxies contain stellar populations with ages <~10Myr and metallicities <=0.15Z_{sun}_. We use photoionization modeling to compare emission lines from models of single-age bursts of star formation to observed emission lines and find that the single-age bursts do not reproduce high-ionization lines including [OIII] or very-high-ionization lines like HeII or OIV]. Photoionization modeling using the stellar populations fit from the UV continuum similarly is not capable of reproducing the very-high-energy emission lines. We add a blackbody to the stellar populations fit from the UV continuum to model the necessary high-energy photons to reproduce the very-high-ionization lines of HeII and OIV]. We find that we need a blackbody of 80000K and ~45%-55% of the luminosity from the blackbody and young stellar population to reproduce the very-high-ionization lines while simultaneously reproducing the low-, intermediate-, and high-ionization emission lines. Our self-consistent model of the ionizing spectra of two nearby EELGs indicates the presence of a previously unaccounted-for source of hard ionizing photons in reionization analogs.