All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly.

What really has to be proved is that $\pi$ is not the root of an algebraic equation with rational coefficients whose degree is a power of $2$. Lindemann proves it for *all* degrees, which is much much more than what is needed for the classical construction problem.

It is not too hard to prove that $\pi$ is not the root of a quadratic equation, and perhaps some clever induction argument could carry the day. The point is that a direct proof for degree $2^n$ has not only never been published, it does not even seem to have been noticed that this much less powerful result is all that is needed.