You will probably also hear many people in America say "I've got a flu" or "I've got the flu", but you would not hear them say "I've got 'a' influenza" or "I've got 'the' influenza". Influenza is the full disease name so people would say "I've got influenza".
Google says in the UK common usage is I've got backache and the USA common usage is I've got a backache... So your book must of been written in the UK.
Google also says minor ailments are countable for example a cold, so that might explain why a flu is commonly used in the USA because you get it and recover over and over again where as once you have cancer it's generally a life thing unless you go into remission and then get a different type of cancer.
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Because it’s grammatically correct