(Abridged) We present a systematic study of the X-ray and multiwavelength properties of a sample of 17 highly radio-loud quasars (HRLQs) at z > 4 with sensitive X-ray coverage from new Chandra and archival Chandra, XMM-Newton, and Swift observations. Eight of the new and archival observations are reported in this work for the first time. New Chandra observations of two moderately radio-loud and highly optically luminous quasars at z > 4 are also reported. Our HRLQ sample represents the top ~5% of radio-loud quasars in terms of radio loudness. We found that our HRLQs have an X-ray emission enhancement over HRLQs at lower redshifts (by a typical factor of ~3), and this effect, after controlling for several factors which may introduce biases, has been solidly estimated to be significant at the 3-4 sigma level. HRLQs at z=3-4 are also found to have a similar X-ray emission enhancement over z < 3 HRLQs, which supports further the robustness of our results. We discuss models for the X-ray enhancements origin including a fractional contribution from inverse Compton scattering of cosmic microwave background photons. No strong correlations are found between the relative X-ray brightness and optical/UV emission-line rest-frame equivalent widths (REWs) for radio-loud quasars. However, the line REWs are positively correlated with radio loudness, which suggests that relativistic jets make a negligible contribution to the optical/UV continua of these HRLQs (contrary to the case where the emission lines are diluted by the relativistically boosted continuum). Our HRLQs are generally consistent with the known anti-correlation between radio loudness and X-ray power-law photon index. We also found that the two moderately radio-loud quasars appear to have the hardest X-ray spectra among our objects, suggesting that intrinsic X-ray absorption (N_H~10^23 cm^-2) may be present.
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