Does Jammit require YUI Compressor or the Google Closure Compiler to be installed on the production server in order to work?
No, it doesn't. It just requires the JVM (or equivalent, such as Open-JDK).
Jammit depends on two third-party Gems: one for Google Closure and one for YUI Compressor. Both Gems are shipped with the corresponding compressor package bundled within the Gem.