  We present an X-ray and multiwavelength study of 33 weak emission-line
quasars (WLQs) and 18 quasars that are analogs of the extreme WLQ, PHL 1811, at
z ~ 0.5-2.9. New Chandra 1.5-9.5 ks exploratory observations were obtained for
32 objects while the others have archival X-ray observations. Significant
fractions of these luminous type 1 quasars are distinctly X-ray weak compared
to typical quasars, including 16 (48%) of the WLQs and 17 (94%) of the PHL 1811
analogs with average X-ray weakness factors of 17 and 39, respectively. We
measure a relatively hard ($\Gamma=1.16_{-0.32}^{+0.37}$) effective power-law
photon index for a stack of the X-ray weak subsample, suggesting X-ray
absorption, and spectral analysis of one PHL 1811 analog, J1521+5202, also
indicates significant intrinsic X-ray absorption. We compare composite SDSS
spectra for the X-ray weak and X-ray normal populations and find several
optical-UV tracers of X-ray weakness; e.g., Fe II rest-frame equivalent width
and relative color. We describe how orientation effects under our previously
proposed "shielding-gas" scenario can likely unify the X-ray weak and X-ray
normal populations. We suggest that the shielding gas may naturally be
understood as a geometrically thick inner accretion disk that shields the broad
line region from the ionizing continuum. If WLQs and PHL 1811 analogs have very
high Eddington ratios, the inner disk could be significantly puffed up (e.g., a
slim disk). Shielding of the broad emission-line region by a geometrically
thick disk may have a significant role in setting the broad distributions of C
IV rest-frame equivalent width and blueshift for quasars more generally.
