Let $\mathcal O$ be an order in an imaginary quadratic field $K$.

Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is not a square?

Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is squarefree and not equal to $1$?

**Is there an elementary solution for 1. and 2.?**

Note that if there was a simple proof of the first statement, then we could perhaps simplify the proof of the integrality of the $j$-invariant at $CM$ points by avoiding reduction to the case of the maximal order.