Your students will sing through illness if you ask them to.
That’s why it’s imperative to ask them not to. I say, “I care more about your vocal health across your lifetime than I do about you singing tonight.” I say, “Don’t ever hurt yourself for the sake of a rehearsal.”
Are there moments when you have to sing or talk through illness? Yes, of course. I am teaching through bronchitis, and my recovery is going more slowly than I’d like – at least in part because I’m talking to classes while vocally fatigued. And even so, I don’t do any vocal modeling when I’m sick. I probably could get some notes out if I wanted to…but at what cost?
There is plenty you can do in rehearsal with music and a pencil, without singing a note. Teach your students how to do that, and never ask them to sing ill.