>>14144101That's one way to read it. The word eligible is a bit hazy in this context. All men under 26 are required to register for the draft. Failing to do so results in a lifelong punishment that cannot be ameliorated after you turn 26. What Biden might have in mind is that if you don't get the injection, you're not eligible to register for the draft, which then opens you up to the lifelong punishments that already are law.
Failure to register is a felony that not only comes with fines up to $250,000 and five years in prison but also a lifetime bans from federal jobs (and most state jobs), student financial aid, federal job training, US citizenship (non-citizen residents are required to register), and security clearances. Once you turn 26, you cannot register to have these lifelong punishments removed.
