I've owned Jeeps before. An XJ a CJ and now a YJ. I don't usually mind all the funny noises they make, but there is one constant squeak that will not go away. It happens every time I go over the smallest bump. It's not from the doors or hard top. It's coming from the driver's side, towards the front I think. Something in the suspension I'm guessing. High pitched and sounds like metal rubbing against metal.
Check the sway bar link bushings, spring eye bushings, shackle bushings...basically check all your bushings, I would think you have one or more that are dry rotted and in need of replacement. If you aren't ready to replace them spray them with some silicone lubricant, I have had varying success with that working to quiet things down until I could replace mine.
I had this problem, it sounds EXACTLY like what you are describing. It drove me crazy figuring it out, until I finally observed that the sheetmetal on the tub that one of my body mount bolts fastens to was partially broken out. Every time I hit a bump, the cracked sheetmetal would scrape against each other causing an ungodly screech. On my Jeep it was the body mount below the passenger side floorboard. I ended up taking the bolt and bushing out, reinforced the broken out mount with heavier steel, and reassembled. Fixed the problem.
My springs are the culprit on mine. I get the same bump-squeak as you describe. Mine magically vanishes for a while every Spring. That's when I do my annual undercarriage painting. After I paint the springs, the squeak will go away for a few weeks; I guess the leaves rub and squeak.