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I may be horribly mistaken but I don't think a square and circle cross-section comparison is effective. You said you made the square with the same moment of inertia which is only half of what's calculating the stress.
max bending stress = M*c/I
Because you made the square the same moment of inertia, I'd assume the c value (or furthest point from the neutral axis) is now shorter (I'm calculating something like 0.438" side to the square?). So your analysis will be a little bit lower because you are calculating closer to the neutral axis than with the circle. However, there will be in increase in the normal stress due to the decreased area of the square... so maybe that will compensate... I dunno...
Have you tried doing a hand calculation for that critical point to see if its reasonable? (I'd give it a try myself but dimensions are missing)
Of course it could just be that COSMOS is liking your square profile better when it meshes.
EDIT: Miscalculated equivalent square, changed number in text