Only a few JPA 2 features are not supported yet by ObjectDB (but expected to be supported soon). See the issue tracking system (access requires logging in to ObjectDB website) for exact details. Notice that the JPA 2 criteria query API is now fully supported (relevant documentation will be released in a few days).
If your application uses old Hibernate API / extensions (e.g. Session instead of EntityManager, queries with HQL syntax instead of JPQL, etc.) you will have to move to standard JPA code.
But if your application already uses standard JPA code - switching to ObjectDB might be straightforward. Just add objectdb.jar to the classpath and set ObjectDB as the provider in the persistence.xml file.
About the data, currently you will have to migrate it by yourself. This can be done by writing a small JPA program that retrieves entity objects from Hibernate, detaches them and then persists them using ObjectDB. A tool that can automate this process will probably be available in the future.
ObjectDB - Fast Object Database for Java (JPA/JDO)