This is my attempt to give a more motivated version of darij grinberg's argument.
Let's try to work in the greatest possible degree of generality. The equation $X^3 + aX^2 + b X + c$ implies the equations in the roots $\alpha_1,\alpha_2,\alpha_3$,
$$\alpha_1 + \alpha_2 + \alpha_3 = -a,$$
$$\alpha_1\alpha_2+ \alpha_2\alpha_3 +\alpha_1\alpha_3=b,$$
$$\alpha_1\alpha_2\alpha_3 = -c.$$
If we imagine that $a$ and $b$ are fixed and $c$ is varying, we can drop the last equation. Because we have two equations in three variables, we can deduce one equation in any two of the variables, say $\alpha_1$ and $\alpha_2$.
When does that equation force their product to be a square?
Writing $\alpha_3 = -a-\alpha_1-\alpha_2$, the equation in $\alpha_1,\alpha_2$ is
$$\alpha_1\alpha_2 = (\alpha_1+\alpha_2) (a + \alpha_1 + \alpha_2) + b. $$
Thus, whenever $b=a^2/4$, the formula on right side is $(\alpha_1 + \alpha_2 + a/2)^2$ and $\alpha_1\alpha_2$ is forced to be a square if $F$ if $\alpha_1,\alpha_2 \in F$.
