The question is that simple. Is there any known consruction of a differentiable map $\phi \colon U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that sends a closed square $Q\subset U$ to a circle?

Of course, the derivative of $\phi$ at the vertices of $Q$ will not be invertible.

Note that the complex map $z\mapsto z^2$ sends the first quadrant to the upper half-plane. Then it straightens out the right angle.