Drop a little lubricating oil through the hole at the top of the razor, then twist it open halfway and drop some oil through the hole at the bottom of the razor. Then open and close the doors a few times to work the oil through. You can also spray some scrubbing bubbles or the like down the center, work it through and let it sit, rinse with warm water, shake out the excess and follow with the oil application. Hope this helps!
I prefer the oil that is used to lubricate electric clippers. Such as WAHL clipper oil, can find it at Wal-Mart.
I found the thread where it shows a step by step procedure for disassembling a Fatboy. Just out of curiosity and to get a look inside, I did only the first step where you remove the little cap from the tip which releases the spring and the silo door assembly which extends down through the handle. The insides of the razor were as clean as a whistle. The squeaking appears to be coming from the little piece in the lower handle shown in the picture below. (The picture was borrowed from the disassembly thread, it's not my actual razor.) I put several drops of clipper oil into all of the interior threads and worked the mechanism for several minutes to work the oil in. I then ran a Q-tip through the handle and over the lower interior threads to remove the excess oil so the whole inside has a fine coat of oil on it. I reassembled it and everything functions exactly like it should and the squeak is still there. I guess it's one of those things that will either work itself out or that I will have to live with. I'm not seeing any apparent damage or wear to the inside from what I could see.