Integrate $\int_{\partial G}(x,y,z) \times N dS$ for $G \subset \mathbb{R}^3$ with a smooth, regular boundary.
$N$ the outward-pointing normal to $\partial G$, and the integral is evaluated per element of the vector.
I'm sure I should use the divergence theorem somewhere here, but I'm not exactly sure how to apply it. For example, I looked at the first entry in the cross product vector and got $yn_z - zn_y$ where $N = (n_x,n_y,n_z)$, but I'm not sure how to continue from here. I feel like I'm missing something... Any help would be appreciated!
Let $\mathbf{A}$ be an answer (we know, that the answer is a vector).
Let us calculate $A_x$. You have written correctly, that:
$$ A_x=\int_{\partial G}(yn_z-zn_y)dS=\int_{\partial G}(\mathbf{f}\cdot \mathbf{N})dS. $$ Here $\mathbf{f}=(0,-z,y)$. Now we may use the divergence theorem for the latter integral. $\mathrm{div}\mathbf{f}=0$, so the integral is $0$, and so $A_x=0$.
We may perform the same trick for $A_y$ and $A_z$. We will obtain $A_x=A_y=A_z=0$. So the answer is $\mathbf{A}=0$.
If we assume that the task may be solved, but the region $G$ is not specified, not that many opportunities remain. $\mathbf{A}=0$ is one of them.