His comparison is kinda bullshit though. In the military such a thing isn't a mandate, it's an order. Willfully disobeying a lawful order makes you subject to Uniform Code of Military Justice. IOW, the law, and it's very different from civilian law. Guarantee you these Marines weren't just "kicked out". They were subject to all sorts of harsh and punitive penalties, not to mention outright derision and harassment before they were discharged.
The process would not have been quick or pleasant, and depending on the type of discharge, they can have problems that follow them into civilian life. The other thing is, most civilian jobs don't require you to go through an extended period of hellish conditions to qualify for said job, and most civvy jobs don't demand the level of perfection and discipline. Refusing a lawful order in any branch is a very big deal, and getting the boot because of it is even bigger. Even if you're in a non-combat arms role and had a relatively short 6-8 weeks of basic followed by a school where there's no real risk to life or health, you've still got a lot invested.
You'd really have to have one hell of a reason for doing it. Have a friend locally who just quit her nursing job of 20+ years over not getting the shot. She just packed her shit, said adios motherfuckers, and was done with the whole process in a short afternoon. Nowhere near the same for people getting the boot from the service. But, Palmer is a tool so ignorance is to be expected.