NEET is "Not [in] Education, Employment, [or] Training"
Shut-in just means you're "shut-in[side]" your room, or someone who almost never leaves the house to socialise or talk to people or do whatever. As the other commenter mentioned, though, it's also used when someone can't leave their home for whatever reason.
Hikikomori is kind of the same thing but it's a Japanese term used to refer to people who never leave their home but still (somehow) function.