Just my derivation. For an arbitrary constant vector ##\vec{n}## we have
$$\vec{\nabla} \cdot [(\vec{n} \cdot \vec{x}) \vec{j}]=(\vec{n} \cdot \vec{x}) \vec{\nabla} \cdot \vec{j} + \vec{n} \cdot \vec{j}.$$
Now integrate this over the whole space and assume that ##\vec{j}## goes to 0 quickly enough at infinity (or most realistically that it has compact support). Then the left-hand side vanishes, because it's a divergence, and thus can be transformed to a surface integral, which vanishes at infinity, if ##\vec{j}## vanishes quickly enough. This implies
$$\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \; \vec{n} \cdot \vec{j} = -\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \; (\vec{n} \cdot \vec{x} ) \vec{\nabla} \cdot \vec{j}.$$
Now use for ##\vec{n}## the three basis vectors of a cartesian reference frame, and you see that the vector equation stated by Jackson holds.
It's independent of whether ##\vec{j}## is a solenoidal field or not. Of course in the case of stationary currents it must be one, but the integral identity is generally valid.