This only hit me when I was driving through Turkey's Cappadocia region: town after town without a single town square. We were driving and trying to find where these towns center were located (we'd guess a town square), but we simply failed to find one.
Even in Istanbul there are no really important squares -- like Taksim square is located in an important area, but it does not have the role of the of squares elsewhere in Europe or in the Americas.
More examples: in Yogyakarta (city in Java, Muslim-majority Indonesian Island), people hang out in the sidewalks instead of squares, because there aren't any. Same for Japan: towns have parks, but no squares.
Does anyone know why there are no squares outside the "west", as in the main place where where town folks socialize?
Ancient Rome cities had a Forum, the area where people socialized and may be analogous to a town square. But Ancient Rome extended way over Turkey, where I could not find town squares.
It's really odd, because from Canada to Argentina, and in Europe, every small town has a square with a bar and church, and it's where people socialize.
(Ha! I can finally talk like a real /r/AskHistorians denizen!)
While we wait for a answer to your exact question, I encourage you to check /u/Mutxarra's answer to "It seems to me that the core of medieval and early modern towns in continental Western Europe was a square (plaza, market, open space, whatever), while in Britain it was a high street. Is this documented or am I imagining it? And if it's the former, then why the divergence?". I think the tl;dr applies to you as well:
[...] it's not as clear cut as Continental western Europe going for squares while Britain went for streets. In fact, regardless of what I told you, you could come to my hometown and think that one of our "modern" squares was the center of the medieval and Early modern town when it was not, so it can also be a matter of foreign perception.