After replacing my PS4 stock hdd, I'd like to place it in a USB bracket and use it for my Xbox One as external storage. Are there any known compatibility issues, or will it work just like any other hdd after reformatting it for Xbox One use?
It ultimately depends on the drivers the USB bracket uses, but take the fact that it is a PS4 hard drive out of the equation - it's irrelevant. They're normally a 3rd party drive in the PS4, not Sony unique.
Basically, yes, it should work.