I found the problem! Here's the issue. Instructions said to copy the libmysql.dll file to the ruby home directory. The file that needs to be copied is NOT the .dll -- it's the libmysql.lib file (or both). I copied that over and it compiled fine. Yay!
$ gem install mysql2 -v 0.3.14 -- '--with-mysql-dir="C:\MySQL-Connector-C-6.1"'
Temporarily enhancing PATH to include DevKit...
Building native extensions. This could take a while...
Successfully installed mysql2-0.3.14
1 gem installed
Installing ri documentation for mysql2-0.3.14...
Installing RDoc documentation for mysql2-0.3.14...