An EU trademark provides protection in all the countries of the European Union based on a single application. If you operate in a large part of the EU, this may be an attractive option. A downside of an EU trademark is the all-or-nothing rule. There are also risks involved in obtaining, maintaining and enforcing an EU trademark. You can apply for an EU trademark at the European Union Intellectual Property Office (EUIPO). You will require the services of a trademark agent to file your application.
EU trademark via an international application
An EU trademark can be obtained as part of an international application. This is a way of securing protection in both the whole EU and in other countries (up to over 110 countries). An upside to this approach is that it minimises the risks associated with the all-or-nothing rule which applies to EU trademarks. International trademark applications must be filed with the office of the country of origin (in your case most likely with CaribIE via BOIP). More about international trademark registration
If within six months of filing your first application, e.g. a Caribbean Netherlands application, you want protection in other countries, you can claim what is called 'priority'. This means that the protection afforded by your subsequent applications (whether for national, Benelux, international or EU trademarks) will then run from the same date as your first one (the filing date). You can claim priority for all countries that are a party to either the Paris Convention or a member of the World Trade Organization. In practical terms, this means nearly every country in the world.