I am a woman but I object to the term Man Flu also. I hate any stereotypes as everyone is individual, I have seem many a woman stay off work with a cold claiming it is the flu. I think it is more about the way you were brought up than about any difference in the genders.
I am glad to hear someone with such a sensible attitude to colds and absence from work. Why wipe out the entire workforce by coming in with a really heavy cold? As you yourself know with asthma it can cause really nasty breathing problems with even a fairly mild cold, and a heavy one can bring on a life threatening astma attack in vulnerable people. I know the truth of this, my mother would be hospitalised twice a year regularly with severe and life threatening asthma attacks triggered by a cold virus turning to bronchitis.