Let us consider the following two properties (cases) of a prime $p$ that has the form $4k+1$ and a natural (not necessarily square-free) number $n<p$:
$2$ is a square modulo $p$: If $n$ is a square modulo $p$, and none of the prime factors of $n$ is of the form $4k+1$, and no pair of prime factors of $n$ (not necessarily distinct) are both of the form $4k−1$, then $p$ is a square modulo $n$
$2$ is not a square modulo $p$: If $n$ is a square modulo $p$, $n$ is odd, and none of the prime factors of $n$ is of the form $4k+1$, and no pair of prime factors of $n$ (not necessarily distinct) are both of the form $4k−1$, then $p$ is a square modulo $n$.
Althought I searched for indicators (including proofs) that these properties are well-known, however, I could not find concrete/established material.
My question: Are these properties known or embodied in existing theorems? I would be greatful for any hints or directions to investigate.
In the case that there are no references to this yet, how we can approach a proof (I am not asking for a complete proof here at all - I would be very grateful for an impulse into the right direction).