I was most certain that the clicking noise I heard was in the bottom bracket. But when I was off of the bike, holding the bike with one hand on the saddle and one hand on the handlebar with front brake pulled, one foot on the ground, and one foot on the right side pedal, when I alternately pushed down on the pedal with my foot and released the pressure, I could hear the click but it was not coming from the bottom bracket after all. (Sorry for the extremely long sentence.)
The clicking sound was a rear rack bolt at a dropout that was not tight enough. It was pretty tight, but pushing down on the pedal flexes the frame, and that frame flex allowed the bottom of the rack to slip and move due to the bolt that was somewhat tight but not really tight enough.
In my case it was a click, not a squeak. But my point is that when you are on the bike, you really can't tell that well where odd noises are coming from.