[Mariner]

The entire gist of the article was for POTUS to remove the waiver requirement and direct all personnel to get the shot.

Does that sound like they have a “right” to decline it to you?

They do not.

[Captain Walker]

Can the president order these men and women to the desert to take up a position in the vicinity of an atomic bomb test?

You don't sign away your humanity once you join the military; there are still lawful and unlawful orders.