From Connecticut Walk Book West, p. 15: "Camp and cook fires are prohibited along the AT throughout Connecticut."
From the Connecticut Forest and Park Association's website, ctwoodland.org/node/136: "Campfires are not permitted along the Blue-Blazed Hiking Trails except at designated sites with permission of the landowner." "Landowner" might include a town or city, I expect. The safe bet in CT is to assume on the AT and the blue-blazed trail system that fires are illegal.