I don't think people from CT have a name. The media sometimes refers to them as Nutmeggers... I've heard the state referred to as "The p*ss stop between New York and Boston", "The Bermuda Rectangle", "The Constipation State", and of course, Hartford referred to as "America's Filing Cabinet" and "The Insurance Capital".
Funny thing about Connecticut. In between Boston-New York and no accents. It's like the two even out here
Let me know the first time someone says this to you. I've never heard it anywhere else. You'll say something like "I love vanilla ice cream". Someone born and raised here will say "So Don't I"... I swear to you. I've heard that from quite a few people here.