I noticed there are two varieties of Maxwell's equations and although I am not sure which form Maxwell originally worked with and perhaps the differential form came much latter, I was wondering if he made use of Green's Theorem. I ask because after reading about Green , I got the impression he was motivated to develop his theorem for trying to understand electrical behavior. I looked at Wiki and James Clerk Maxwell lived from 1831 to 1879 and George Green from 1793 to 1841. Certainly Maxwell was old enough to have read about Green's work before Green passed away.
Of course Maxwell knew Green's theorem, by the time he was writing this was the common knowledge. Maxwell's book has a mathematical preliminary chapter (chapter 2) where he explains mathematical tools he uses, and this contains Gauss, Green, Stokes theorems and much more. (In fact he anticipates what was later called Hodge theory). In the chapter where Maxwell equations are derived, (Chap 9, vol. 2) he freely passes from differential to integral formulations and back.
Green's book (1828) remained in obscurity for some time, until William Thomson (future Lord Kelvin) found it in 1845 and made his results widely known. And yes, Green's book was called "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism", so it is about electricity and magnetism.