  We report the results from Chandra and XMM-Newton observations of a sample of
10 type 1 quasars selected to have unusual UV emission-line properties (weak
and blueshifted high-ionization lines; strong UV Fe emission) similar to those
of PHL 1811, a confirmed intrinsically X-ray weak quasar. These quasars were
identified by the Sloan Digital Sky Survey at high redshift (z~2.2); eight are
radio quiet while two are radio intermediate. All of the radio-quiet PHL 1811
analogs are notably X-ray weak by a mean factor of ~13. These sources lack
broad absorption lines and have blue UV/optical continua, suggesting they are
intrinsically X-ray weak. However, their average X-ray spectrum appears to be
harder than those of typical quasars, which may indicate the presence of heavy
intrinsic X-ray absorption. Our radio-quiet PHL 1811 analogs support a
connection between an X-ray weak spectral energy distribution and PHL 1811-like
UV emission lines; this connection provides an economical way to identify X-ray
weak type 1 quasars. The fraction of radio-quiet PHL 1811 analogs in the
radio-quiet quasar population is estimated to be < 1.2%. We have investigated
correlations between relative X-ray brightness and UV emission-line properties
for a sample combining radio-quiet PHL 1811 analogs, PHL 1811, and typical type
1 quasars. These correlation analyses suggest that PHL 1811 analogs may have
extreme wind-dominated broad emission-line regions. Observationally,
radio-quiet PHL 1811 analogs appear to be a subset (~30%) of radio-quiet
weak-line quasars. The existence of a subset of quasars in which
high-ionization "shielding gas" covers most of the BELR, but little more than
the BELR, could potentially unify the PHL 1811 analogs and WLQs. The two
radio-intermediate PHL 1811 analogs are X-ray bright. One of them appears to
have jet-dominated X-ray emission, while the nature of the other remains
unclear.
