A relation $\rho$ is not reflexive if there exists some $a\in A$ which is not related to itself, i.e. $\exists a:a\not\rho a$. $\rho^2$ means the relation applied twice.
Is there a direct proof of this claim?
If a relation $R$ on $A$ is not reflexive, $R^2$ on $A$ is also not reflexive.
I can think of some counterexamples. Assume there is one such and let this be $a_1$.
Then $a_1\rho a_1 = a_2$. Since $a_2\ne a_1$, $a_2$ must be reflexive.
$\implies a_2 \rho a_2$ is reflexive.
$\implies a_1 \rho^2a_1$ is reflexive.
Hence we have found a case where a relation $R$ where the relation is not reflexive is true but "$R^2$ is also not reflexive" is false.
Hence the claim is false.