Is there a way that I can still code in eclipse (without using Maven) and still be able to have jenkins run my builds? Also, I've been having a hard time getting it to do anything with my builds.
Yes, but is there a way to have the project set up so I can keep the default New -> Java Project or does it HAVE to be in Maven?
If you're referring to the standard directory structure (src/main/java), it's merely a convention and not a technical requirement. You can do without it. Of course, you might have to change how you refer to your dependencies (including Bukkit), but that is the whole point of using Maven - it takes care of downloading and managing each dependency to the point of automating the build process. It's what makes tools such as Jenkins possible in the first place. You could look into using Ant, or perhaps even calling javac yourself in a batch script. But this will probably mean more work overall compared to Maven, especially the batch solution ...