Q:

durch oder über den Marktplatz, die Fußgängerzone

I would use durch for both die Fußgängerzone and der Marktplatz, but in a text in my textbook, 

über den Marktplatz 

was used. Which one is correct then?

A:

A Platz (town square) is the surface of a larger area on the ground. You can not be in a town square. This would mean, that half of you body is stuck in the ground. You can be auf (on) a Platz. You just can walk through the air above the square, but not through the square itself.
A Zone (zone) is a region with some specific rules/laws. So you can be in a zone and you can walk through it.

A:

Durch means you are inside the item and pass through it while über means you cross it.

Ich bin durch das Gewimmel über den Marktplatz gegangen.

I crossed the market square through the masses.
It gets interesting as soon both options are valid:

Das Schiff fuhr durch den Ärmelkanal.

The ship passed the English Channel. (It's a ship going east-west.)

Das Schiff fuhr über den Ärmelkanal.

The ship crossed the English Channel. (It's a ferry, Dover-Calais.)

