Is my 16x2 LCD display broken?
I have one of those and when I apply 5v to it coming off from 3.3v all of the characters that are written there become white squares just like it happens with your LCD, the solution is to go with a small screwdriver to the backside of it and dial down the brightness until you can see the characters again, maybe try supply it with less voltage.

P.S. I know you said you tried turning the dial down but please check first that you're not outputting squares to the LCD by loading a simple "hello world" style program and going from there. Don't forget to turn everything off for a few seconds and then ON again, also, there's always the possibility that it's your MCU that isn't doing things right, if you have a second MCU that you know for a fact to be OK, use it to test the LCD. Don't forget to check your connections.