We present Chandra observations of six bright quasars that represent extrema in quasar emission-line properties --- three quasars each with small and large blueshifts of the CIV emission line with respect to the systemic redshift of the quasars. Supplemented with seven archival Chandra observations of quasars that met our selection criteria, we investigate the origin of this emission-line phenomenon in the general context of the structure of quasars. We find that the quasars with the largest CIV blueshifts show evidence, from joint-spectral fitting, for intrinsic X-ray absorption (N_H~10^22 cm^-2). Given the lack of accompanying CIV absorption, this gas is likely to be highly ionized, and may be identified with the shielding gas in the disk-wind paradigm. Furthermore, we find evidence for a correlation of Delta(g-i), the relative continuum color in the observed-frame optical, with the hardness of the X-ray continuum, and an analysis of independent Bright Quasar Survey data from the literature supports this conclusion. This result points to intrinsically red quasars having systematically flatter hard X-ray continua without evidence for X-ray absorption. We speculate on the origins of these correlations of X-ray properties with both CIV blueshift and Delta(g-i) and discuss the implications for models of quasar structure.