Let $p\in(0,\,1]$ . In this paper, the authors prove that a sublinear operator $T$ (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces $H^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ to some quasi-Banach space $\mathcal{B}$ if and only if $T$ maps all $(p,\,2,\,s_{1},\,s_{2})$ -atoms into uniformly bounded elements of $\mathcal{B}$ . Here $s_{1}\ge\lfloor n(1/p-1)\rfloor$ and $s_{2}\ge\lfloor m(1/p-1)\rfloor$ . As usual, $\lfloor n(1/p-1)\rfloor$ denotes the maximal integer no more than $n(1/p-1)$ . Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space $L^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $p>1$ or the Hardy space $H^{p}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $p\le1$ but near 1 to the Lebesgue space $L^{q}(\mbi{R}^{n}\times\mbi{R}^{m})$ with some $q>1$ .