Will a .22 cal. necksizing die work on all .22 cal. cases?

It should.

Yes, with reservations. In centrefire 22s there are two bullet diametres .223 & .224 mainly associated with 22 Hornets which have been made in each of these depending on date of manufacture and some other early 20th century firearm. Just take care that the N/S die is suitable for the bullet you are using.

There are several variations of neck thickness used in .22 BR and .22 PPC dies.
A short fat die such a .22 BR with the correct dia neck will neck size about anything.

On the other a .220 Swift die will work only on itself as does the .22 Hornet.