So it looks like you are downloading a pre-compiled gem.
I'm not sure how much of this you understand but the mysql gem is
different from most ruby gems in that it is a compiled interface
implements a ruby interface to the mysqlclient API.
My guess is that the mysql....so file that ships with your gem is not
correct for your system. I'm a little fuzzy here on the finer points
of the mswin32 and 64 bit environments but I do know that there are
several ways to get a miss-match.
On my system, Mac OS X, I start from the source and compile the gem as
part of the install step. This reduces the likelyhood of a miss-
matched executable but causes more work preparing for the compile.
It's the preparation part that appeared to be broken for me but I've
figured out how to fix that and built a functioning gem.
Do you have a c compiler on your system? If so, have you ever used it
and do you want to give compiling your gem a shot?