2006 Toyota Camry 108K miles. It still has factory clutch. Since two nights ago, when I step on the clutch pedal and press it down, it has begun to make mild squeaking noise that’s kind of sounds like when you are rubbing two hard rubber surfaces. This squeak happens whether the engine is running or not, and it’s only when the clutch pedal is going down and not when it’s coming back up. Since the same squeaking noise happens with engine off, I suspect it’s probably not the clutch disc. The noise is under the dashboard in front of the firewall where my left foot is when sitting inside the car. What could be the reason for this squeaking? To start the process of elimination, what do you advise I lube? What kind of lube do I need? Thanks.
Look for a pivot point on the clutch pedal and use a little light weight oil on it and see if that helps.
A small squirt of silicone lubricant on the lever pivot points should do the trick.
When my old pickup got up in age I had to lube the clutch lever pivot points every few years. It really was no more than an extremely minor inconvenience.
I sprayed the silicone lube on the spring and moving parts of the clutch pedal under the dashboard yesterday, and so far the squeaking seems to have stopped when the clutch pedal is pressed. Back in January, a mechanic did say my clutch pedal feels hard. Is it likely during the past 11 years, the clutch pedal moving parts have gone dry and that is what caused the squeaking to being three days ago? Clutch pedal also feels more smooth when I press it so hopefully clutch itself has not yet worn out enough to warrant replacing it?
Yup.
Dry pedal linkage has no effect whatsoever on clutch wear. Only one wear in your ankle bones. 
Try changing out the bushings. I bought new ones for my 1979 Celica’s clutch pedal, as I was having the same problem you describe. Lubed new bushings with lithium grease. No more squeaks.
On my 1984 Chevy Cavalier and my 1999 Honda Civic, one area that squeaked was where the cable ended, or the slave cylinder was located, on the outside of the clutch housing where the pedal action gets transformed into the lever that acts on the clutch throwout bearing.
On the Cavalier, the squeaking pivot point was on the outside of the clutch housing, easy to see behind the radiator. I dribbled some light oil - maybe Triflow or Sturmey Archer oil - a couple or few times, and as it made its way down, the squeak/creak went away.
On the Civic, it’s a similar location, but under a rubber boot, and hard to see or feel exactly where to lubricate. I have sprayed Triflow into the area under the boot and that has also taken care of it, at least for a few years, when I have to repeat the treatment. I’m careful not to overdo it, for fear of oiling the clutch friction surfaces. Close to 179,000 miles and the Honda’s clutch is still doing fine.