We have searched for redshifted neutral hydrogen 21 cm absorption towa
rd sources from the Stickel et al. ''red quasar'' subsample. The red q
uasar subsample is taken from the 1 Jy sample of flat-spectrum radio s
ources and is comprised of the 15 sources that are undetected on the P
OSS. Five of these red quasars have been searched for redshifted H I 2
1 cm absorption to optical depth levels of a few percent, and four sho
w strong absorption, with neutral hydrogen column densities between 4
and 80 x 10(18) x (T-s/f) cm(-2). This 80 % success rate for the red q
uasars compares to the much lower success rate of only 11 % for detect
ing H I 21 cm absorption associated with optically selected Mg II abso
rption line systems. The large neutral hydrogen column densities seen
toward the Stickel et al. red quasars provide circumstantial evidence
supporting the dust-reddening hypothesis, as opposed to an intrinsical
ly red spectrum for the AGN emission mechanism. The lower limits to re
st-frame values of A, are between two and seven, leading to lower limi
ts to the spin temperatures for the neutral hydrogen between 50 K and
1000 K, assuming a Galactic dust-to-gas ratio. We consider the questio
n of biases in optically selected samples of quasars caused by dust ob
scuration. Overall, the data on the red quasar subsample support the m
odels of Fall & Pei for dust obscuration by damped Ly alpha absorption
line systems and suggest that (1) there may be a significant, but not
dominant, population of quasars missing from optically selected sampl
es because of dust obscuration, perhaps as many as 20 % at the POSS li
mit for an optical sample with a redshift distribution similar to the
1 Jy flat-spectrum quasar sample, and (2) optically selected samples m
ay miss about one-half the high column density quasar absorption line
systems. The redshifted H I 21 cm absorption line detections presented
herein are toward the sources 0108 + 388 at z = 0.6685, 0500 + 019 at
z = 0.5846, and 1504 + 377 at z = 0.6733. No absorption is seen towar
d 2149 + 056 at z = 0.740 at a level below that seen for the three det
ections, although there is some uncertainty in this case as to the exp
ected line redshift. In some systems, the absorbing gas is in the vici
nity of the AGN, as either circumnuclear material or material in the g
eneral ISM of the AGN's host galaxy, as is probably the case for 0108
+ 388 and 1504 + 377, and in other systems the absorption is by gas as
sociated with galaxies cosmologically distributed along the line of si
ght to the quasar, as may be the case for 0500 + 019. The Westerbork S
ynthesis Radio Telescope spectrum of 1504 + 377 confirms the lack of H
I 21 cm absorption associated with the narrow molecular absorption li
ne system at z = 0.67150.