In http://mathproofs.blogspot.com/2005/07/mapping-square-to-circle.html, there is a derivation of the mapping from a unit square to a unit circle. Looking up wikipedia tells me the canonical form of an ellipse is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

and the aforementioned article starts with:

$\frac{(x^')^2}{x^2} + \frac{(y^')^2}{b^2} = 1$

and proceeds to solve for b.

My question is why is it that $x^2$ is the correct value for $a^2$ in this particular problem?

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He wants to take any vertical line (line of constant x) into the primed coordinate system so that it provides a piece of the ellipse. Eventually he wants the image of $[-1,1] \times \{-1\} \cup \{1\} \times [-1,1] \cup [-1,1] \times \{1\} \cup \{-1\} \times [-1,1]$ to be the unit circle in the primed coordinates. If you look at the final result, if $x=\pm 1$ or $y=\pm 1$, $x'^2+y'^2=1$.