For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true?

Claim: If $k$ is prime and $n!+k$ is a square, then $n \le k$.

Proof: Clearly $k$ can't be $2$ (mod $4$ considerations), so $k$ is odd and then, by mod $4$ considerations, is $1$ mod $4$. Then $n!+k = m^2$ implies that for each odd $p \le n$, $(\frac{k}{p}) = 1$, which implies $(\frac{p}{k}) = 1$ by quadratic reciprocity (since $k$ is $1$ mod $4$). Also, $n!+k = m^2$ directly implies $k$ is $1$ mod $8$, so $(\frac{2}{k}) = 1$. Therefore, if $n \ge k$, then each $p \le k$ has $(\frac{p}{k}) = 1$, and thus by multiplicativity, we get $(\frac{m}{k}) = 1$ for each $m \le k$, which is impossible, since there are $\frac{k+1}{2}$ quadratic residues mod $k$. $\square$

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WE Tutorial School's answer below shows that $n \le k$ if $k$ is a nonsquare composite. The argument is as follows. Note $k \mid \frac{n!}{k}$, since if $k = rs$ for $1 < r < s < k$, then $r,s$ appear in $n!$ as well as $k$ (assuming $n > k$). So $\frac{n!}{k}+1$ is relatively prime to $k$, but then that $k(\frac{n!}{k}+1) = n!+k$ is a perfect square means that $k$ must be a perfect square, which we assumed it isn't.

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This leaves open the case of $k$ a perfect square. It should be noted that answering question (1) in the affirmative would then improve Dabrowski's result, so seems hard.

In addition to mathworker21's work, we have this claim: if $k$ is a non-square composite number such that $n!+k$ is a perfect square for some positive integer $n$, then $n\le k$. We are left with the case where $k$ is a perfect square.

Since $k$ is a composite number which is non-square, $k=rs$ for some integers $r$ and $s$ such that $1<r<s<k$. If $1<k<n$, then $$n!+k=k\left(\frac{n!}{k}+1\right)=k\ell,$$ where $$\ell=\frac{n!}{k}+1=k\left(\frac{n!}{rsk}\right)+1=k\,(r-1)!\left(\frac{(s-1)!}{r!}\right)\left(\frac{(k-1)!}{s!}\right)\left(\frac{n!}{k!}\right)+1$$ is clearly an integer coprime to $k$. However, as $k\ell$ is a perfect square with $\gcd(k,\ell)=1$, it follows that both $k$ and $\ell$ are perfect square, but this contradicts the assumption that $k$ is non-square.

If $1<k<n$ and $k=t^2$ for some positive integer $t>1$, then we need to show that $$\ell=\frac{n!}{k}+1=\frac{n!}{t^2}+1$$ is not a perfect square. Equivalently, we need to show that $\frac{n!}{t^2}+1$ is never a perfect square of an integer for any positive integer $t$ such that $1<t<\sqrt{n}$. I am not sure how to do that, but it can be easily seen that for $\ell$ to be a perfect square, $n>16$ so that $t,2t,3t,4t< n$, and $$\ell = t^2\left(24\ (t-1)!\ \frac{(2t-1)!}{t!}\ \frac{(3t-1)!}{(2t)!}\ \frac{(4t-1)!}{(3t)!}\ \frac{n!}{(4t)!}\right)+1.$$ I am not quite sure what to do with that.

However, as for when $n!+1$ is a perfect square, this is known as Brocard's problem. So far, the only known values of $n$ that work are $n\in\{4,5,7\}$.

Let $n$ and $k$ be non-negative integers such that $n!-k$ is a perfect square. We want to show that either $(n,k)\in\big\{(0,0),(1,0),(0,1),(1,1),(2,1),(2,2),(3,2)\big\}$ or $k\ge n$.

One can see that $n!$ is a perfect square if and only if $n=0$ or $n=1$. One can see that $n!-1$ is a perfect square if and only if $n=0$, $n=1$, or $n=2$. One can easily see that $n!-2$ is a perfect square if and only if $n=2$ or $n=3$. We assume from now on that $k>2$. Suppose for the sake of contradiction that $k<n$.

We can use the same reasoning as my work above to establish that $k$ cannot be a non-square composite number. However, it is also easily seen that $n\geq 6$. Hence, if $k$ is a perfect square, then $$\frac{n!-k}{k}=\frac{n!}{k}-1\equiv -1\pmod{4}$$ so $\frac{n!-k}{k}$ can never be a perfect square. Hence, in this situation, we are left with the case where $k$ is prime. However, mathworker21's argument can be used again (all credits go to him, so I am making this post a community wiki post).

Suppose now that $k$ is an odd prime. Using the same argument, $-k$ is a quadratic residue modulo $p$ for every odd prime natural number $p\leq n$. By quadratic reciprocity, if $p<k$, then $$\left(\frac{p}{k}\right)\left(\frac{k}{p}\right)=(-1)^{\frac{(k-1)(p-1)}{4}}.$$ As $k\equiv -1\pmod{8}$, we conclude that $$(-1)^{\frac{(k-1)(p-1)}{4}}=(-1)^{\frac{(p-1)}{2}}=\left(\frac{-1}{p}\right).$$ That is, $$\left(\frac{p}{k}\right)=\left(\frac{k}{p}\right)\left(\frac{-1}{p}\right)=\left(\frac{-k}{p}\right)=1.$$ Finally, $$\left(\frac{2}{k}\right)=(-1)^{\frac{k^2-1}{8}}=1.$$ Therefore, every positive integer less than $k$ is a quadratic residue modulo $k$, and this is a contradiction.