An Equilateral on a Parabola

Triangle $DEF$ has vertex $E$ on the $x$-axis and $F\left(\dfrac{1}{4}, 0\right)$, with $|DF| = |EF|$, and the midpoint $M$ of side $DE$ lies on the $y$-axis. Let $\Gamma$ be the locus of $D$.

1. Find the equation of $\Gamma$. 2. An equilateral triangle $ABC$ has all three vertices on $\Gamma$, and the line $AB$ has inclination $45^\circ$. Find $|AB|$.