Define a function $$\mathscr R: \mathbb N \to \mathbb N, \ \ \mathscr R(n) = \lceil \sqrt{n} \rceil ^2 -n.$$ IE. the distance of $n$ to the next prefect square. Sequence A068527 on OEIS.
If $\mathscr R(n) = 0$, $n$ is a perfect square. If $\mathscr R(n) > 0$, $n$ is not a perfect square. For example, $\mathscr R(3)=1$, for $3+1=4=2^2$.
Now, assume we apply $\mathscr R$ multiple times. What happens?
If $\mathscr R(\mathscr R(n)) = 0$, the distance of $n$ to the next square is a square. And again, $\mathscr R(\mathscr R(n))$ can be greater than zero. For example, $\mathscr R(1271)=25=5^2$ and $\mathscr R(\mathscr R(1271))=0.$ Thus $1271 = 36^2-5^2$ and $1271 = 31*41.$
Can we apply $\mathscr R$ until we get zero for all numbers? The answer is no, for if $\mathscr R^k(n)=2$ for some $k$, we can go no further, for $\mathscr R(2)=2$ and the result is infinite loop. Assume we stop calculation to $2$ in such cases.
Two examples. $$\mathscr R(87)=13,\ \mathscr R(13)=3,\ \mathscr R(3)=1,\ \mathscr R(1)=0$$ $$\mathscr R(93)=7,\ \mathscr R(7)=2,\ \mathscr R(2)=2,\ \mathscr R(2)=2\ \dots$$
I have done some tests on a computer with this process and I would like to ask about something I have noticed. Let $K_n$ be the smallest number of iterations needed to get to either $0$ or $2$, starting from $n$. I have been unable to find a counterexample to the statement: $$\forall n \in \mathbb N, \ \ K_n \le 8.$$
For every number, for which $\mathscr R^{K_n}(n) = 0,$ we can now give a representation $$n = (n+\mathscr R(n))-(\mathscr R(n)+\mathscr R(\mathscr R(n))-\mathscr R(\mathscr R(n)))\dots$$ Notice that $\mathscr R^k(n)+\mathscr R^{k+1}(n)$ is a perfect square. Putting this is a bit clearer form, $$n=a^2-b^2+c^2-d^2\dots,$$ as long as the iteration stops at $0$.
Given my observation and the last statement, is it possible to represent $n$ as an alternating sum of squares, where the first term is $\lceil \sqrt{n} \rceil ^2$, of maximum of $8$ terms?
$\forall x \in \mathbb{N}, K_n \leq 8$ is false. If $K_n=8$ and $\mathscr R (m)=n$ then $K_m=9$.
On the other hand, the 8 alternating squares should be possible, considering Lagrange's four-square theorem, the only problem being your restriction on the first square, but I believe the negative squares can fix that.