God cannot lie and God cannot change.
James 1:17 Every good and perfect gift is from above, coming down from the Father of the heavenly lights, who does not change like shifting shadows.
Hebrews 6:17 Because God wanted to make the unchanging nature of his purpose very clear to the heirs of what was promised, he confirmed it with an oath. 18God did this so that, by two unchangeable things in which it is impossible for God to lie, we who have fled to take hold of the hope offered to us may be greatly encouraged.
I am not talking about what God can do in some vague and meaningless sense. The proposition you challenged was "Whether God can do whatever he wants," not "Whether God can do anything whatsoever in some abstract or speculative sense."
You were hitting the buzzer over what you thought was an error of logic. Surely you see the logical difference between "God can do anything whatsoever," and "God can do whatever he wants."
So, likewise, the quote from James only applies if we know that God wants to change, and the one from Hebrews only if we know God wants to lie.
In a larger sense though, Protestants generally seem to think, contra Paul, that Freedom is the ability to chose whatever course of action one pleases. Catholics, with Paul, understand that freedom is the ability to choose the good and to act on one's choice.
We see a difference between the alcoholic's craving for a drink and our Lady's grace-given desire to be the handmaid of the Lord. The first is not freedom.
The relevance is that God is utterly free For him to want to change or to lie is a failure or defect of wanting, not true wanting. He knows the good, 'wants' the good, and does it. He can do whatever he wants, as the Psalmist says.
It is, I must admit, a source of pleasure to have someone contradict the plain sense of Scripture to prove a Catholic wrong.