I think the problem with the whole "what doesn't kill you makes you stronger" idea that your coworkers seem to be espousing is that a serious viral infection (or other serious illness or injury) isn't like going hard at the gym, it's like getting hit by a car. A lot of people end up with a weakened immune system for months to years following a major illness, leaving them more susceptible to death from flu or some other infection, or just more likely to get smaller sicknesses. That's why flu vaccines help prevent secondary pneumonia in people who would have been weakened by the flu, or how the measles vaccine helped prevent deaths in other illnesses in children, because it turns out that kids who get measles are more likely to get seriously ill from something else in the following year. (
Article with some sources)
I say this as someone who got viral pericarditis 3 years ago, and the recovery process to semi-normal took months, and I'm still not fully where I was before I got sick. If I could have been vaccinated for that virus (it was probably Epstein-Barr, the one that causes mono), I would have gladly taken feeling bad for a day from the vaccination instead of what I got- a couple of multi-day hospital visits and months of being too weak to walk a mile. Recovery has been a looooooong road.