I don’t think this is true. It’s just that if someone catches the flu and their immune system deals with it quickly, the person may be unaware that they even had the flu.
This happens frequently when encountering a strain of rhinovirus, say, that you’ve already encountered, but I can see no reason why this shouldn’t sometimes happen on encountering a new virus. And a stronger immune system would mean greater probability of early victory.
I think some posters above have given good responses, mainly citing those viruses that can cause severe sickness or death via an overresponse of the immune system.
If I may now hijack to more difficult examples that are interesting to think about:
AIDS – In the early stages, a strong immune system may help the virus spread more quickly, since it infects immune cells. But later on, it’s basically how many T-cells you have remaining that dictates your life expectancy, so you may still be better off overall starting with a higher count.
Prion diseases (which are of course infectious) – There is some evidence that the immune system plays some role here, and can help break up plaques. It’s not certain. But those that die from vCJD are typically young, implying (just implying) that the strength of the immune system is not the critical factor.