If b is negative, define c = -b, and use the linked result to show that

a^2 - b^2 = a^2 - (-c)^2 = a^2 - c^2 = (a + c)(a - c) = (a + b)(a - b)

You can do this if you extend your concepts of length and area to signed quantities. You have to be clear to explain how the setup of the drawing works, but instead of cutting away a square of side b out of the corner of the square of side a, you might end up appending a negatively-signed square of side −b.

There’s no need for this extension: https://news.ycombinator.com/item?id=42425681

Well the whole point here is that we're doing geometry. If you just want to do symbol twiddling you don't need the picture at all.

I am not convinced since in this world you can have the following rects:

1 x 1

1 x -1

-1 x -1

Is just the middle one a cut out?

I reckon you need the axiom from the field to do generalize this to negative numbers.

https://en.m.wikipedia.org/wiki/Field_(mathematics)