It may be illegal to strike but it's not illegal to to not work if you're not paid is it?

Depends on the job. There are some jobs where refusal to do your job may carry significant legal penalties (imagine if airline traffic controllers walked out with planes in the air)

Sure. I'm assuming there's some level of resolution that serves as a transaction. Like maybe a day, or a shift. In your example, I'm talking about your worker not walking out on a flight in final approach, but rather notifying the employer after the shift that she will not be showing up for any future shifts until she is paid.

I don't know the specific rules regarding calling in sick etc for that industry. I know most companies have policies regarding absences, and once you hit a certain point, you can be terminated with cause. (presumably just refusing to come in would be treated worse than just calling in sick excessively) Not only would that make them ineligible for unemployment, but I assume that would impact someone in that industry pretty negatively. (How many different employers are there for ATCs?)

I'm not suggesting that anyone call in sick. That would be a lie. And I'm not suggesting that the stopping of work is permanent like quitting. This situation is one in which the employer is temporarily not paying, therefore there shouldn't be any problem with the employee temporarily not working.

How exactly is being legally forced to come in to a job, especially one that isn't even paying you, not either slavery or at least indentured servitude?

It's only that if you are not allowed to quit. And while this is somewhat pedantic, they are being "paid", but the release of that pay hasn't been authorized. Their situation is total shit to be certain, but comparisons to bona fide slavery are dramatic.

It's a system in dire need of reform, no doubt. I don't agree with the way it works either.

Break the law? They aren’t slaves. Fuck the law.

Nobody is going to jail for quitting their job.

Quitting and striking are different. Sure you could quit, but then when the shutdown ends, you won't have a job at all.

I'm sure some people may consider that option, but others may be hoping to ride it out. Remember, some people want to stay in a federal job due to retirement plans, etc. so they are thinking beyond the short term effects.

This is why many are calling out "sick". I saw an estimate of 10% in some cases. It's not quite as effective as a strike, but it can still make a statement at least.