  Highly radio-loud quasars (HRLQs; $\log R>2.5$) at $z\gtrsim 4$ show apparent
enhanced X-ray emission compared to matched HRLQs at lower redshifts, perhaps
due to a redshift-dependent fractional contribution to the X-ray luminosity
from inverse-Compton scattering of cosmic microwave background photons
(IC/CMB). Using new {\it Chandra} observations and archival X-ray data, we
investigate this phenomenon with an optically flux-limited sample of 41 HRLQs
at $z = 4$--5.5 all with sensitive X-ray coverage, the largest sample utilized
to date by a wide margin. X-ray enhancements are assessed using
X-ray-to-optical flux ratios and spectral energy distributions. We confirm the
presence of X-ray enhancements at a 4.9--5.3$\sigma$ significance level,
finding that the median factor of enhancement is $\approx 1.8$ at our sample
median redshift of $z\approx 4.4$. Under a fractional IC/CMB model, the
expected enhancement at lower redshifts is modest; e.g., $\approx 4$% at
$z\approx 1.5$. We also investigate a sample of seven radio-loud quasars (RLQs;
$\log R>1$) at even higher redshifts of $z=5.6$--6.8, using new and archival
X-ray data. These RLQs also show evidence for X-ray enhancements by a median
factor of $\approx 2.7$ at a 3.7--4.9$\sigma$ significance level. The X-ray
spectral and other properties of these $z=5.6$--6.8 RLQs, however, pose
challenges for a straightforward fractional IC/CMB interpretation of their
enhancements.
