Are there any Halachik or Hashkafic issues with using rogaine or some other hair loss prevention medicine?
It is clear from this answer of R. Elchanan Lewis, that there is no problem:
It is a standard treatment for men and is not a problem of Beged Isha.
From what I read on their website I can't see any Halachic problem applying the treatment apart from the issue of pulling out hair, therefore if one can't apply the Rogaine without pulling out hair that may be a reason for refraining from its usage on Shabbat.
If you can avoid the treatment by applying right before and straight after Shabbat, that will be better.
From the discussion it is evident that sans Shabbat issues, there is no problem.