The Verdict: Not eligible
You do not need a work permit: as an EU, EEA or Swiss citizen you can take a job or work for yourself in Croatia freely. If you stay over three months, register your residence with the local police administration.
Free movement means no employer has to do anything on your behalf. Registration gives you a registration certificate; after that, sort out health insurance with the Croatian fund and your tax position.
Five years of continuous residence gives permanent residence. If you have Croatian ancestry, look at our Citizenship by Descent and Diaspora & Religious Return quizzes, which can reach citizenship without residence.