The Polynomial That Divides Its Own Composition

Let $f$ be a nonconstant polynomial with positive integer coefficients. Prove that for every positive integer $n$,

$n \mid f(n) \implies n \mid f(f(n) + 1).$

Wait — prove the stronger claim from Putnam 2007 B-1:

Let $f$ be a nonconstant polynomial with positive integer coefficients. Prove that for any positive integer $n$, $f(n)$ divides $f(f(n)+1)$ if and only if $n = 1$.