The corners of a geometrically perfect square should have no width. But if they have no width they don't exist. Therefore the corners must have a width. If they have a width they can be looked at as additional sides. look at this picture
Why not also argue that a perfect three-dimensional body is not conceivable, because it needs to be bounded by a surface, which has zero width and therefore does not exist? If the surface does not exist, then the body also cannot.
That's somewhat similar to Zeno's argument that there cannot exist many things.
I admit that the human mind can get entangled into certain "problems" when thinking about perfect geometric forms. If we take away the corner points of a perfect square, say, remove the single points (0, 0), (1, 0), ... of the unit square, have the square's corners gone blunt or is it still perfect? Or more fundamentally, does it still have corners?
But this can at least be mathematically handled so is it really inconceivable?
And even if it is, can we make the step "inconceivable" => "impossible"? This inference has a long tradition but is not unequivocally accepted.
The question is somewhat biased on perception.
The corners by themselves have no significance. Corners are formed only when two line intersects. Now if we are talking about geometry any corner do not have a width because it's not an object by itself but just a name given to identify a point in space.
Now if we talk about perception, by definition a dot is a small round mark or spot. However if you see a dot or a circle on your computer screen is it actually round? The answer is that you perceive those things as round but in fact they are made out of small square shaped pixels.
Therefore we can say that corners do not have width according to geometry and by perception if we go to say like it got 0.00000000001 width which in itself got no significance
Perfection and existence are different issues. A boundary can exist as the junction of two other things which exist, even though it has no independent physical existence.
But your question title asks something rather different. A perfect square cannot physically exist. The reason for this is not any problem with the existence of its boundary, rather it is in the imperfection of its boundary. Due to the finite size of atoms, quantum fluctuations and other phenomena, no physical object can perfectly align with a mathematical square.
Plato proposed an independent world of ideals, in which the perfect square did exist, but it rather begs the question as to what we mean by existence.