In other words, \[a+b+c \mid a^{2n}+b^{2n}+(a+b)^{2n}-n(a^2b^2+(a^2+b^2)(a+b)^2).\]But letting $c \to \infty$ here, we see that this happens iff \[a^{2n}+b^{2n}+(a+b)^{2n}=n(a^2b^2+(a^2+b^2)(a+b)^2)\]for all distinct positive integers $a,b$. But then of course this must already be a polynomial identity in $a,b$. In particular, comparing degrees on both sides we must have $n=2$. But for $n=2$, the identity is indeed true!