The square is defined as a union of lines, and the line integral of a whole shape is the summation of that of the curves it is a union of with open intervals no

The square is defined as a union of lines, and the line integral of a whole shape is the summation of that of the curves it is a union of with open intervals not including the "corners" (points of 'infinite curvature' as you say). I think this definition would suffice because we just need to show that the sum is closed over cutting, no?