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I disagree.
You're implying that square circles are platonic concepts that aren't empirically verifiable. I would argue that they entirely are, just the task would be too difficult to be worth anyone's while. Just because something is in someone else's field of perception, doesn't make it any less real if it's a particular hypothesised shape, or a particular hypothesised colour. I could simply ask someone if they've seen a square circle and if everyone says no, can comfortably believe there aren't any till I perhaps see one, just as if I ask about black swans and if everyone says no, comfortable b eleive they don't exist unless I see one.
This is the assumption made in your last paragraph and I completely disagree. I've frequently found that things I thought were impossible happened. That kind of dogmatic certainty sounds awefully dangerous. While thinking about logic in that kind of self-consistent, but externally inconsistent sense seems to be absurd. One can describe a particular mythology that might make sense in a self-consistent way, but when related to other systems of belief isn't coherent.
Yes, any real-world circle is imperfect and deviates from being a circle in the mathematical sense. Something that's square deviates a lot from the circle in the mathematical sense and is thus no real circle.