Solution:

Let $a$ be any number in the list; we wish to prove that $a$ is a square. Let $b$, $c$, $d$ be any three other elements of the list, all different from each other. Then, the given implies that there are integers $p$, $q$, $r$ with $a b c = p^{2}$, $a b d = q^{2}$, $a c d = r^{2}$. Multiplying these together gives $a^{3} b^{2} c^{2} d^{2} = p^{2} q^{2} r^{2}$; equivalently,
$$a = \frac{p^{2} q^{2} r^{2}}{a^{2} b^{2} c^{2} d^{2}} = \left(\frac{p q r}{a b c d}\right)^{2}.$$
So $a$ is the square of a rational number, but since it is an integer, we conclude that it is the square of an integer, as needed.