Federal law [0] says you cant be fired for taking up to 12 weeks off per year to take care of, among other things, "major illness" (understood to mean that it does not cover things like colds or flu, unless they require hospitalization). But, it also only applies to businesses with over 50 employees, so non-chain restaurants are unlikely to follow this law.
[0] https://en.wikipedia.org/wiki/Family_and_Medical_Leave_Act_o...
Other places simply have it for everyone, and a union negotiation won't take away the safety net, as it is a right. It can only add benefits to the minimum.
https://www.workplacefairness.org/paid-sick-leave
I haven't reviewed them all, but the ones I've looked at are pretty meager, ~40 hours a year (and often conditional on employer size), which won't do much other than take the edge off if you get more than a little sick.
The most populous state in the country requires a minimum of 3 days paid sick leave for employees: https://www.dir.ca.gov/dlse/Paid_Sick_Leave.htm
Employers cant legally prevent employees from using this time. But, given that low-income people have little ability to fight violations of the law, I'd hardly be surprised to hear that people are forced (or feel forced) to work while sick.