= Solution
Take the outward <orientation of a surface>. The <divergence> is $\nabla\cdot\mathbf F=y^2+z^2+x^2$, so the <divergence theorem> converts the closed <surface flux> into an integral over the box:
$$
\int_S\mathbf F\cdot d\mathbf S=\int_{-a}^a\int_{-b}^b\int_{-c}^c(x^2+y^2+z^2)\,dz\,dy\,dx.
$$
Each term separates; for example the $x^2$ term is $(2a^3/3)(2b)(2c)$. Summing gives
$$
\boxed{\int_S\mathbf F\cdot d\mathbf S=\frac{8abc}3(a^2+b^2+c^2).}
$$
Here $a,b,c>0$ are the box half-lengths. Reversing <orientation of a surface> would negate the result.
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