How can I make my bed non-box-spring frame stop squeaking?
The frame is squeaking because two pieces of wood are rubbing against eachother.  The first thing I would do is start tightening any screws, nuts and bolts on the bed frame.  If this does not improve it you will need to locate where the squeak is coming from and determine if you can do something to hold the two components together better, such as driving a screw through the affected area.