Middleborough is the name of the school that Be More Chill takes place in.[1]
It is not explicitly mentioned in the official cast recording[2], but in the audios taken of the live 2015 production, it's stated that "It has been a tough week here at Middleborough" by Mr. Reyes during The Play scene.
In the 2018 live Off-Broadway show, there is a podium during the I Love Play Rehearsal scene that shows a "Middleborough Wombats" logo in green font. This alludes that the school team's name is the Wombats, and the school colours are green.
In "The Bathroom Intervention", when Micheal confronts Jeremy, at around 1:59 he states "How did they end up in a highschool?? In New Jersey?" explaining the school is somewhere in NJ.
Also in "The Play" it's briefly mentioned by the teacher "Do you think I wanted to teach highschool drama? In New Jersey?"