Repair squeaky MDF board for a Ottoman Divan Bed
MDF is particularly prone to sagging, and the only way to flatten it and prevent future sagging is to a fix a  brace or batten to it. 

A 1x2" piece of hardwood, as long as possible, should be used (multiple, if possible, along the width). The best way to fasten it would be with Screws through the MDF top (make sure to drill pilot holes and countersinks to prevent warping the mdf).

Im not sure about the squeaking... But if to from the top contacting the sides when you're on top, a bit of foam tape or weatherstripping in the right spot can help that.