  Using the \prospector\ spectral energy distribution (SED) fitting code, we
analyze the properties of 19 Extreme Emission Line Galaxies (EELGs) identified
in the bluest composite SED in the \zfourge\ survey at $2.5 \leq z \leq 4$.
\prospector\ includes a physical model for nebular emission and returns
probability distributions for stellar mass, stellar metallicity, dust
attenuation, and nonparametric star formation history (SFH). The EELGs show
evidence for a starburst in the most recent 50 Myr, with the median EELG having
a specific star formation rate (sSFR) of 4.6 Gyr$^{-1}$ and forming 15\% of its
mass in this short time. For a sample of more typical star-forming galaxies
(SFGs) at the same redshifts, the median SFG has a sSFR of 1.1 Gyr$^{-1}$ and
forms only $4\%$ of its mass in the last 50 Myr. We find that virtually all of
our EELGs have rising SFHs, while most of our SFGs do not. From our analysis,
we hypothesize that many, if not most, star-forming galaxies at $z \geq 2.5$
undergo an extreme H$\beta$+$[\hbox{{\rm O}\kern 0.1em{\sc iii}}]$ emission
line phase early in their lifetimes. In a companion paper, we obtain
spectroscopic confirmation of the EELGs as part of our {\sc MOSEL} survey. In
the future, explorations of uncertainties in modeling the UV slope for galaxies
at $z>2$ are needed to better constrain their properties, e.g. stellar
metallicities.
