As you alluded to, the Euthyphro Dilemma is pretty much all one needs to know to refute this typical Christian apologetic.
If morality is independent of God and exists outside of Him, then it means God is not all powerful and that He is subject to something greater than himself, ie, objective morality. No Christian can accept this proposition and still believe God is omnipotent.
However, if the source of all morality is from God and is not something outside and independent of God then objective morality does not exist, since it is only a reflection of Gods divine commands. It would mean morality is arbitrary and that might is right after all, as Satan claimed.
It's an intractable dilemma.
Some Christians have said the dilemma is false, claiming there is a third option. But it's not very convincing.