No. Consider $x'=g(x,y)$, $y'=h(x,y)$, with $g(x,y)=2|x|^{1/2}$ for $y=x^2$ and $g=0$ otherwise and similarly $h=4|x|^{3/2}$ on $y=x^2$ and $h=0$ otherwise. Since each ray intersects the parabola at most once, the antics on $y=x^2$ don't affect directional differentiability.

We can check directly that $x=t^2$, $y=t^4$, $t\ge 0$, and $x=y=0$ are solutions with initial value $(0,0)$.