How do you uninstall a kitchen sink sprayer hose and install a new one if the old connector just spins around in its housing?

If the hose-end just spins, it's probably retained by a "circlip." This is a wire made in the shape of a three-quarter circle. You'll need to us a pliers (needle-nose if you have one) to remove the circlip and then the hose-end will just slip off.