Approaching Another Lab With A Project Proposal Based On My Thesis Chapter
Is there an ethics/legal problem with taking the idea to another professor to see if they are interested in having another student complete the work...?


Your current collaborator/PhD supervisor is not returning your emails. This is a signal that work will not proceed. For confirmation, you can outright ask them whether they want to proceed, and inform them that you'll proceed alone if you do not receive a response within a certain window, e.g., two weeks. Assuming you still don't receive a response, you can work with who you want to complete the work. (I'm unsure why another student is being sought. You'll surely need to contribute for any collaboration to be viable.)


  ...or does the original university own the rights to the data that the idea was based on?


With regards rights, you haven't given enough information to answer. If the prior work appears in your thesis, then anyone can continue the work. If the prior work is whatever you conceived whilst employed by the university, then rights will be defined in your contract. If the prior work includes a dataset owned by the university, then permission to use that dataset will be required.