"Anxious" for "eager" was a pet peeve of my grandfather's (I was only 3 when he died, so though I do remember a few things about him, this thing I know only through my father).
I tend to eschew "wrong" or "right" when talking about word usage, but I think "anxious" for "eager" works better when anxiety is part of the equation. For instance, I might be "anxious" to go somewhere if I am running late, but if I'm talking about my hypothetical upcoming Hawaiian vacation, I'd be eager not anxious.
In short, it helps if you understand the words you use.