Can shootouts happen in the town square?
Yes, you can have Shoot-Outs in Town Square.

On page 18: 


  As a Noon play, one of your unbooted dudes can choose a dude controlled by another player at the same location, and call out that dude (challenge that dude to a shootout). This doesn’t boot your dude, and so as long as your dude stays unbooted, they can call out an opposing dude each and every time it’s your turn to make a play." 


And on page 12 it states:


  There are three types of in-town locations. Each deed is its own location. Each outfit’s home is a location (with a few special rules). And the area in the middle of the table is a location called the town square.


Shootouts happen when dudes are at the same location and there are three types of in-town locations, one of which is the town square.