Groups with order divisible by $d$ and no element of order $d$
Solution 1:
Let $d$ be your composite, squarefree number. Let $p$ be the largest prime dividing $d$. Then the symmetric group $S_p$ will have order a multiple of $d$ (since its order is $p$-factorial, a multiple of every prime up to $p$), but no element of order $d$. Every element of $S_p$ is a product of disjoint cycles; to have an element of order a multiple of $p$, one of those cycles must be a $p$-cycle; but there's no cycle disjoint from that $p$-cycle.