Bob, I am guessing that there is no drawer slide hardware and the drawer bottoms just ride on the face frame rails. What happens is that the wood on the bottom of the drawer wears away over time and use but not in a straight line. There will be a high spot next to the drawer front. When you close the drawer, you get to the high spot and the squeek occurs.
One way to cure this is to level the bottom of the drawer side by sanding and then glue a thin slat of 3/4" wood on to bring the drawer side back to the correct width. A better way is to do the same as above but leave the slat about .030" less thick than it should be. Glue this on and then get some of the 1/2" wide Nylo-Tape from Rockler or similar tape and put it on the bottom of the slat. This tape will let the drawer slide easily and, since there is less friction, the drawer bottom will not wear. This is item # 70615 at www.Rockler.com and it costs $5.99 for a 10 foot roll. I had to do this to my old wood desk and it works great now.