This came to my mind after reading some introduction to maximum entropy probability distributions.
Independence can be derived from the following four assumptions:
(1) average momentum of particles inside the box is fixed at 0
(2) average kinetic energy of particles inside the box is fixed
(3) the gas velocity distribution must be maximum entropy probability distribution under the above two constraints (1) and (2).
(4) kinetic energy of a particle can be expressed as $f(v_x) + f(v_y) + f(v_z)$ for some function $f$. (not true for relativistic particles)
(Hopefully (2) and (3) are somewhat more plausible-looking than independence. (2) seems related to equipartition theorem or is it enough to say that the average kinetic energy is fixed simply because of law of large numbers? And (3) is mostly asserting that there is no other hidden constraints (other than (1) and (2)) to be discovered.)
To derive independence, suppose $p(v_x,v_y,v_z)$ is a probability density function (on velocity) that does NOT have independence between x, y, z components of speed. We want to compare the entropy of $p$ to that of $p'$ where $p'$ is defined as $p'(v_x, v_y, v_z) = p_1(v_x) p_2(v_y) p_3(v_z)$ where $p_1$ is the marginal distribution of x component of velocity from $p$ and similarly for $p_2, p_3$. If $p$ shows average momentum of 0, and average kinetic energy of 1, then the same is true for $p'$. ($p'$ and $p$ having same average kinetic energy is derived from (4)). But it is a result in information theory that $h(p) < h(p_1) + h(p_2) + h(p_3) = h(p')$ (where $h(p)$ denotes entropy of $p$). This means that $p$ CANNOT be maximum entropy probability distribution under constraints (1) and (2).