  We summarize panchromatic Extragalactic Background Light data to place upper
limits on the integrated near-infrared surface brightness (SB) that may come
from Population III stars and possible accretion disks around their
stellar-mass black holes (BHs) in the epoch of First Light, broadly taken from
z$\simeq$7-17. Theoretical predictions and recent near-infrared power-spectra
provide tighter constraints on their sky-signal. We outline the physical
properties of zero metallicity Population III stars from MESA stellar evolution
models through helium-depletion and of BH accretion disks at z$\gtrsim$7. We
assume that second-generation non-zero metallicity stars can form at higher
multiplicity, so that BH accretion disks may be fed by Roche-lobe overflow from
lower-mass companions. We use these near-infrared SB constraints to calculate
the number of caustic transits behind lensing clusters that the James Webb
Space Telescope and the next generation ground-based telescopes may observe for
both Population III stars and their BH accretion disks. Typical caustic
magnifications can be $\mu$$\simeq$10$^4$-10$^5$, with rise times of hours and
decline times of $\lesssim$1 year for cluster transverse velocities of
$v_{T}$$\lesssim$1000 km s$^{-1}$. Microlensing by intracluster medium objects
can modify transit magnifications, but lengthen visibility times. Depending on
BH masses, accretion-disk radii and feeding efficiencies, stellar-mass BH
accretion-disk caustic transits could outnumber those from Population III
stars. To observe Population III caustic transits directly may require to
monitor 3-30 lensing clusters to AB$\lesssim$29 mag over a decade.
