The top came off easily, just a couple of springy catches at the back. Top was moving in contact with the front. I put foam tape between them and tightened the screws holding the front. Also th screws now visible holding the sides to the counter. The squeek is gone.
I've used a product called 'Slip-Not', for a bunch of years. I think it's also sold for under area rugs as well.
Cut squares of it to go between items that rattle. Stuff is so good that back when we smoked, we could leave an ash tray on the table while driving down the road. Now it just goes under Lap Top, etc., on the dinette. It's also washable in warm water with the laundry.
Still haven't quite conquered the oven, and stove top, burners, cover, however. Three or four layers helps the cover, but the oven racks must be 'loaded' to even start to muffle them. Of course I still need to replace the shocks, so that's part of it. I have 23K on original shocks, so I know they are weak.
2011 Dodge 1500 C'boy Caddy
2000 Jayco C 28' Ford chassis w V-10 E450
Doghouse 36' or so Trophy Classic TT