Migrate JupyterHub app state to new k8s cluster? (AWS EKS)
Depending on your deployment, I would copy all yaml files responsible for deploying the JupyterHub and copy the data from Persistent Volumes.
You could do that by using kubectl cp this is nicely explain in this post How to copy files from kubernetes Pods to local system.
It all depends on how you deployed the JupyterHub, you need to provide more details regarding how was that deployed and what is exactly being used and how.