Direct proof that $\pi$ is not constructible
I've just read in the book The Number $\pi$ by Eymard and Lafon that no such proof is known.
“The proof that it is impossible to square the circle does not involve direct demonstration of the non-constructibility of the number $\pi$. As far as we are aware, it is not known how to do this! One proves that it is not algebraic, which is much more restrictive, then one uses the fact that a constructible number is algebraic.” [§4.2, p. 134]