Can I just share the jMonkeyEngine code and keep my code closed or would I have to make my game open as well? Sorry if this is a noob question.
As per the about page: http://hub.jmonkeyengine.org/introduction/
"Free, open-source software (under the New BSD license) – Use our free engine for commercial, educational, or hobby game development"
BSD license allows for closed source commercial development of work based on that software. So yes, your game can be closed source.