What is the name of the economics free trade "punnett square" I remember doing these in High School.It was a 2x2 square, where you could work out the math to show that countries were better off focusing on their efficiency/competitive advantage, and that free trade made both better off.

Not looking for an analysis of if the logic is right or wrong or whatnot, just what is the name of the square (assuming it has one)
Perhaps you mean an Edgeworth box?
Here is an example demonstrating Ricardo's theory of comparative advantage using an Edgeworth box:

Source: Policonomics 

This shows that through trade both Portugal and England can have more wine and more cloth after opening up trade between the countries. It also shows that even if Portugal is more productive than England at producing both commodities then trade is still improving (that is what makes it comparative and not absolute advantage). 

Alternatively, perhaps you mean a 2x2 table like this which shows the labor needed to produce each good:


Source: KALYAN CITY LIFE

To my knowledge this table has no special name.