Rather general versions of these theorems have been established for "sets of finite perimeter." You can find a recent paper on this subject here:
Chen, Gui-Qiang; Ziemer, William P.; Torres, Monica, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Commun. Pure Appl. Math. 62, No. 2, 242-304 (2009). Available at arXiv:0709.3673 [math.AP]. Zbl 1158.35062.
On the other hand, you say you don't want "heavy machinery," so it is hard to guess what you are looking for.
