jmc

The graph of the parabola is shown below:



Let the coordinates of point $A$ be $(x,y)$. Then since the midpoint of $\overline{AB}$ is the origin, the coordinates of $B$ are $(-x,-y)$. Both of these points must lie on the parabola, so we plug them into the equation for the parabola to get the equations $$\begin{array}{rl}y& =4x^2+7x-1,\\ -y& =4(-x{)}^2+7(-x)-1\Rightarrow y=-4x^2+7x+1.\end{array}$$Setting the two equations equal to eliminate $y$, we have $4x^2+7x-1=-4x^2+7x+1$, or $8x^2=2\Rightarrow x^2=\frac{1}{4}$. So $x=\frac{1}{2}$ (the negative alternative for $x$ gives the same answer) and $y=4{\left(\frac{1}{2}\right)}^2+7\left(\frac{1}{2}\right)-1=\frac{7}{2}$. Thus, point $A$ is at $(1/2,7/2)$ and point $B$ is at $(-1/2,-7/2)$. The length of $\overline{AB}$ is then $\sqrt{{\left(-\frac{1}{2}-\frac{1}{2}\right)}^2+{\left(-\frac{7}{2}-\frac{7}{2}\right)}^2}=\sqrt{50}$. Hence, $A{B}^2=\boxed{50}$.