You need to work on an axiomatic foundation for counting squares.

Very interesting. I applaud the spirit if not the success of the enterprise. Paraphrasing scientist Laurence Kraus, scientists go to work everyday with the hope and intent of proving their colleagues wrong.

I suppose the whole idea of thinking in any field according to Kraus and probably most thinkers past, present and future is in small part an exercise in building and fixing theories, old and new, but in large part consists of taking the wrecking ball to existing paradigms and breaking them.

Do continue this task - you might break math and that would be something very interesting to watch and also very satisfying given that I've always envied the mathematically talented lot.

Addition is different form adding things together. 1+1=2. Even if I place 2 rats in a room and come back a week later and find 20, 1+1 still equals 2. They're completely different unrelated statements.

It depends on what you are adding together. 1+1 does not always equal 2. If your counting cuboids rather than squares, then you can add the cuboids together in such a way that they create 3.. It depends on what your unit is. If my unit is a square or a triangle for that matter then the result stands. Remember before you even get to the point of adding something together you have to have the identity that x=x but with some shapes this breaks down totally like triangles and squares i.e 4={4,5}. If i'm using circles for example then I can't arrange circles in any way that affects x=x no matter how I arrange them. With other shapes you can. All addition is is the act of adding 1 quantity to another and then summing the result. If I'm adding squares, triangles or cuboids then it all goes pear shaped so to speak..Your rats example is a different story. Your units are rats and initially the answer is 2 but this changes over time. With squares and triangles the result is instant and subject to other factors.

Amphiboly. You're adding squares of a type and then including squares of another type.

If I'm counting squares then a squares a square. Squares combine to make other squares. How many squares are there on a chess board? its more than 64. I define a square as a shape with 4 linear sides of equal length. When I add them together I can create more from less. Somethings can be made from parts. A car is made of loads of parts but they still sum to make a car. A square can be made from parts but those parts can be squares :)

You can make things from things.
Squares. Line segments. Points.
Anything if you zoom out far enough.
You just need to have a split perspective.

If the greatest common divisor of a and b is 1 and ab is a square, both a and b are squares.
With the same restrictions, if $ab^2 = c^2, a$ is a square.

The math is interesting, but there's just no philosophical issue here.

If I were you, I'd look at rectangles instead. Maybe have a look at partitions too.