  The ultraviolet (UV) continuum slope
($\beta~$where$~f_\lambda\propto\lambda^\beta$) of galaxies is sensitive to a
variety of properties, from the metallicity and age of the stellar population
to dust attenuation throughout the galaxy. Considerable attention has focused
on identifying reionization-era galaxies with very blue UV slopes ($\beta<-3$).
Not only do such systems provide a signpost of low-metallicity stars, but they
also identify galaxies likely to leak ionizing photons from their HII regions
as such blue UV slopes require the reddening effect of nebular continuum to be
diminished. In this paper we present a search for reionization-era galaxies
with very blue UV colors in recent JWST/NIRCam imaging of the EGS field. We
characterize UV slopes for a large sample of$~z\simeq7-11~$galaxies, finding a
median of$~\beta=-2.0$. Two lower luminosity (M$_{\rm{UV}}\simeq-19.5$) and
lower stellar mass (6-10$\times10^7$M$_\odot$) systems exhibit extremely blue
UV slopes ($\beta=-2.9~$to$~-3.1$) and rest-optical photometry indicating weak
nebular line emission. Each system is very compact (r$_e\lesssim$260pc) with
very high star-formation-rate surface densities. We model the SEDs with a suite
of BEAGLE models with varying levels of ionizing photon escape. The SEDs cannot
be reproduced with our fiducial (f$_{\rm{esc,HII}}$=0) or alpha-enhanced
(Z$_\star<Z_{\rm{ISM}}$) models. The combined blue UV slopes and weak nebular
emission are best-fit by models with significant ionizing photon escape from
HII regions (f$_{\rm{esc,HII}}$=0.5-0.8) and extremely low-metallicity massive
stars (Z$_\star$=0.01-0.06Z$_\odot$). The discovery of these galaxies
highlights the potential for JWST to identify large numbers of candidate Lyman
Continuum leaking galaxies in the reionization-era and suggests low-metallicity
stellar populations may be common in dwarf galaxies at $z>7$.
