Depends who's posting...
Doing literally anything at all will regenerate your immune system. It's replenishing stocks of white blood cells all the time--I read somewhere that they last about a year, compared to a few months for red blood cells.
The story's not actually that sensationalistic. It's just saying that it may be beneficial for e.g. chemo patients to fast as there's some evidence that it encourages the production of more white blood cells.
The question is... does that benefit you more, in your particular situation, than the potential downsides of fasting. I mean most people who advocate fasting as a lifestyle thing are not actively going through chemotherapy or have some other health problem that particularly needs a boost of white blood cells. Is it particularly useful to "regenerate [your] entire immune system"?