You can use a delivery method for specific countries by using the "Regions" function. You can set the delivery areas (for example EU, non EU) and connect the delivery methods to be available only in specific areas.
Set the regions and add countries
First create the regions on the page Settings > Country Settings > Regions. Give a name to the region, for example "EU" and save. Click the new name to open and choose the countries for this region and save.
Connect delivery methods to the regions
When the regions are set you can open the delivery method you want to use on the page Settings > Delivery and payment. If you have not yet created a delivery method it is also possible to do that now. Choose the region you created from the drop down menu of the field "Allow usage for the following delivery addresses" and save. When this is done the delivery method will be available only to customers that have a delivery address in one of the countries for that region.
Note! To be able to use this function the customers must be able to choose the country for their delivery address. This can be set to visible and required on the page Settings > Basket settings > Registration. To add or remove available countries go to page Settings > Country Settings > Countries.