the casing nails used in the installation have worn the nail hole a bit bigger than the nail and the squeak is probably the wood squeaking up and down on the casing nail. It would have been nice to have had these screwed down then recessed and plugged, but oh well.
You can do several temp fixes, one is to spray some water around the nail hole and the water will be absorbed into the hole and cause the wood to swell. if the nails are not galv, which I am sure they are not, then use WD-40. This might be your best bet anyway.
Now if you can get up under the staircase, if you have a closet there, then screw toenail the treads to the supports again. Do not go through the tread.
You can also get some Schwinn oil less grease in a spray can from any bike shop. this stuff is great for this project because no oil finish and works great. this could also lub the holes.