Does the monk's unarmed strike use Dexterity or Strength when making Opportunity Attacks? Our DM swears up and down that it goes off of Strength, but everything we read says that all of a monk's unarmed strikes use Dexterity.
You use your strength modifier. Opportunity attacks involve making a melee basic attack, and the melee basic attack power is defined as using the strength modifier - so that's what you'll be using.
I'm not sure what you've read about making the attack with a dex modifier. Your group might be confused and going by 3.5e rules: in 3.5e, the Monk could use the Weapon Finesse feat to use Dex for attacks. That's not a part of 4e, though. Make sure you're going by 4e rules, and not getting mixed up with 3.5e ones - 4e shares very little with its predecessor.
One of the major changes is that the stat used for an attack is determined by the power used, not the weapon; opportunity attacks usually use the "melee basic attack" power, so that's where you look to find out what stat to use for its attack and damage rolls.
Melee basic attacks can be made with a weapon, and this is where the Monk's Unarmed Strike feature comes in, providing you with the option of a weapon to use:
This "weapon" appears on page 63 of the PHB3; there isn't much more to it.
So what do your unarmed attacks do with Unarmed Strike?
Your attack roll uses your strength modifier, with a +3 proficiency bonus from your Unarmed Strike, and the enhancement bonus of your Ki Focus, if you have one.
If you hit, you roll for: weapon damage (the Unarmed Strike's 1d8), and add your strength modifier and your Ki Focus enhancement bonus (if you're using one).