Is there a better time, dealing with general health, to take a shower?
 In the evening, before bed, or in the morning after walking up?

I doubt there would be a general health concern from varying the time
at when you take a shower, but there are some minor concerns.  Your
hair and skin are very tempermental.  You need to allow proper drying
time before exposing your hair and skin to the elements.  If you have
ever taken a shower on a winter morning and went outside, you can feel
the effect.  In winter, I would say the evening is the best time to
take a shower.  If you are settled in for the night, this is the best
thing to do.  Give your body and hair time to absorb the moisture
before taking it outside.  On a warm day, that is not going to matter
as much.  It's better to have your hair a little wet on a hot day
especially, because the sun will dry your skin and hair out,
otherwise.  Taking a shower on a hot day in the morning, is better for
you.  You also don't want to take a shower directly before bed. 
Moisture caught in your hair could translate into your pillow, and
start to develop mildew.  Then you would have some general health
concerns.

I think that people should take showers every Saturday. It's a good
day to take a shower, not in the morning, not at night, but in the
middle of the day, between your luncheon with the in-laws and dinner
with the wife. Having a handle like 'saiori' makes it seem you are
Japanese. I just learned about your type of people, the water going in
to the pillow, not a chance. It's so bloody hot that it evaporates as
soon as you exit the bath house. Why don't you just follow my advice,
every Saturday, any more than that is sick, too much water, less is
not necessarily bad, just make sure you clean special spots on a
regular basis, the human body tends to catch bacteria in crevasses and
after a while, they will start to smell, then rash. That's not a good
thing. PS - Soap is bad too, it causes chafing.