Do the three hands on an analog clock (hours, minutes, seconds) ever divide the face of the clock into three equal segments, i.e. 120 degrees between each hand?

This is not possible.

The ratio of the difference between the angular velocities of the
second and minute hands is 60 times that of the minute and the hour
hands. So if the integer part of the total angle (in revolutions,
since 12:00:00) between the minute and hour hands is 1/3 or 2/3, the
integer part of the angle between the second and minute hands will be
zero.

In other words, whenever the angle between the hour and minute hands is
120°, the second hand is parallel to the minute hand!