Back in that provincial closeminded backwater known as the Middle West, where I have lived my previous thirty-four years, the word "square" was reserved for shapes that have four sides of equal length and connect at four right angles. The beloved Midwestern burgh where I am a professor (Madison, WI) and the hearty Midwestern hamlet where I went to graduate school (Bloomington, IN) both had a single place referred to as The Square, which consisted of four segments of street of basically equal length and that intersected at right angles.
Meanwhile, if you go jogging in the Cambridge/Somerville/East Cambridge areas, every 10-12 blocks you will pass a sign designating yourself as in or near a Square (I'm not just talking about the major ones--Harvard, Porter, Davis, Inman, Central, Kendall--but all kinds of ones with names of men who sound like either hearty old city officials or celebrated local quarterbacks, including nicknames like "Ram" and "Gus" being part of the square's official name.)*) I've spent time craning my neck about at each of these squares, and every time, not to get all Gertrude Steinish on y'all, there is no square there. At times, what gets called a "square" here out east is just something we would call a "funny intersection" back in Middle America.
* I think it was Gus. There's definitely a R_(firstname)_ "Ram" R_(lastname)_ Square somewhere along one of the routes I've run.