EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" of $\eta(-2)=0$.
OP:
Inspired by this stack exchange post, I attempted to perform an Abel sum for $\eta(-2)$, since I read from Wikipedia that $\eta$ is Abel summable, which I assume implies Abel summable everywhere. By $\eta$ I mean Dirichlet's eta function $$\eta(z)=\sum_{n=0}^{\infty}(-1)^{n-1}n^{-z}$$
I only read about Abel sums last night, and am very much uneducated on the subject of analytic continuation, so I tried to show that $\eta(-2)=0$ without any help by way of practice - I might have succeeded, but my methods are strange and I am myself unsure whether I can do the kind of things I'm doing! My approach was as follows:
$$A\text{-}\eta(-2)=\lim_{x\to1^-}\sum_{n=0}^{\infty}\eta_n\cdot x^n=\lim_{x\to1^-}\sum_{n=0}^{\infty}(-1)^{n-1}n^2\cdot x^n$$ $$A\text{-}\eta(-2)=\lim_{x\to1^-}\left[0+x-4x^2+9x^3\dots\right]=\text{?}$$
I did not know where to go from here, but I looked for a simple derivative or integral trick, as the post I referenced had made.
EDIT: This is the strange part that I would like a formal proof verification of:
$$\int(-1)^{n-1}n^2\cdot x^n\space dn=\frac{(-1)}{n}^n\cdot n^2\cdot x^n-\int\frac{(-1)}{n}^n\cdot 2n\cdot x^n\cdot\ln|x|\space dn=n(-x)^n-2\ln|x|\int(-x)^n\space dn=$$
$$n(-x)^n-2\ln|x|\frac{x^{n+1}}{n+1}$$
Therefore:
$$\lim_{x\to1^-}\sum_{n=0}^{\infty}(-1)^{n-1}n^2\cdot x^n=\lim_{x\to1^-}\frac{d}{dn}\left[\sum_{n=0}^{\infty}n(-x)^n-2\ln|x|\frac{x^{n+1}}{n+1}\right]=\frac{d}{dn}\left[\sum_{n=0}^{\infty}n(-1)^n\right]=\frac{d}{dn}\left[1-1+2-3+4\dots\right]=0\text{?!}$$
which is seemingly correct, although it seems far too strange and I'm sure this is a mistaken pseudo-proof.
Perhaps
$$\frac{d}{dn}\left[\sum_{n=0}^{\infty}n(-1)^n\right]=\sum_{n=0}^{\infty}(-1)^n+n^2(-1)^{n-1}=\eta(-2)+\sum_{n=0}^{\infty}(-1)^n$$ although this is a contradictory nightmare suggesting that $\eta(-2)=\eta(-2)+L$ where $L$ is the rightmost sum, either taken to be divergent or $\frac{1}{2}$, depending on the context! And so I have shown my utter ignorance, since to me this all seems like conjectured nonsense, even though I appreciate the principle and I fail to see an error in my calculations...
Please help! I'm looking for intuition/direct proof for the Abel sum of $\eta(-2)$ and for the errors in my insane and uneducated attempt to tackle this. I'm especially curious as to whether an expression like $\frac{d}{dn}\sum_na_n$ even works, and as to whether my "answer" $\frac{d}{dn}\left[1-1+2-3\dots\right]=0$ is accidentally correct, or pure nonsense.
And please bear in mind when you answer/comment that my "knowledge" of analytic continuation and "super-summing" comes only from two Mathologer YouTube videos :) so my language and statements are probably going to seem very off.