They should, I doubt they will though. Rs in Congress will be annoyed and more along, unless there's a lawsuit (over FOIA requests) and it gets media attention.
If the President had a (D) next to his/her name, it would've been WWIII to conservative media.
In Trump reality, it's thisisfine #560459
I dunno man, I'd have to think that even (or maybe especially) Republicans want to be able to trust their staff. Constitutional issues aside, this is a huge breach of trust and the chain of command for those staffers.