Penalty shots during regulation count toward the save percentage of the goalie (and toward the shooting percentage of the shooter). See for example:
Michael Frolik scored one of four Blackhawk goals on a penalty shot. He has a 1/2 Shooting % and Jimmy Howard has a 24/28 save statistic (not 24/27 as it would be if it didn't count) and 4 GA. You can also see that a penalty shot counts against the "Even Strength Goals Against", at least if it's not during a powerplay separately from the penalty that incurs the shot - Jimmy Howard has 3 EV GA and allowed one powerplay goal to open the scoring to Hossa.
Shootout goals do not count; for example:
Andrew Hammond has a 36/38 save statistic, for .947 sv%, despite allowing 3 goals during the shootout (out of 5...