On the main page I noticed that JJWT is supposed to run on all JDK ...
That is definitely incorrect / inaccurate. The initial commits for the JJWT project were for JDK 1.6. And I can see that the code uses generics, so compiling / running for earlier than JDK 1.5 will be impossible.
The POM.xml file for the latest version of the project sets the source and target version for the project to JDK 1.7. So certainly the artifacts that a standard build will generate now will NOT run on a Java 6 platform. The class file version will be too recent for a Java 6 JVM to understand.
Now you could try changing the JDK level to 1.6 in the POM file in the HEAD version. However, the JDK version for JJWT was bumped past 1.6 four years ago, so there are liable to be build issues if you wind back. (The developers could have introduced Java 7+ dependencies; e.g. using try with resources syntax or Java 7+ APIs.)
A better idea would be to use an old version of JJWT. You should be able to figure out what version you need to use (or build) for JDK 1.6 compatibility by looking at the project's git history on Github. Look at the relevant changes to the POM file and when they happened. I think it is version 0.9.0 or thereabouts.
Of course a better solution would be "unstick" your project. JDK 1.6 is 8 years past its end of life, and 3 years past the end of Oracle's JDK 1.6 (paid) extended support.