Yes, ferret do catch influenza; use precautions like hand washing, face
masks, no kissing, etc.  if you have influenza (or even if you think
that it might be a cold, because even though ferrets don't get human
rhinoviruses they can get some of our sinus infections, or sometimes a
person will be infected with an influenza to which the person has a bit of
immunity so the symptoms will be low enough to seem like just a cold).
 
Ferrets do cough and hack more during shedding seasons so comb them, give
them furball laxative, and even give a bath to wash away loose hairs.  It
is possible to have a blockage from too much ingested fur.
 
Some ferrets do get allergies but it doesn't seem to be many of them.
One common cause seems to be perfume in laundry detergents or softeners
that aren't allergan "free".
 
Finally -- and very importantly -- chest x-rays can be very important when
a ferret has a persistent or pronounced cough.  It is entirely possible
for the ferret's chest to sound normal yet for there to be a serious lung
infection (Have been there.) or -- more commonly -- cardiomyopathy
(Ditto.) which is not at all uncommon in older ferrets.  In the first,
prompt treatment will save the ferret's life.  For the second, prompt
treatment will provide months or in some cases even years of happy and
comfortable time and extra life (instead of suffering the untreated result
of a disease that is very uncomfortable and quite rapidly fatal without
treatment).
[Posted in FML issue 3583]