How to write JUnit tests for a function that is only called when the program is run?

If you want to run a test for the logic of the import itself - it should have nothing to do with GUI at all.

So it all depends on your code - not every code is unit-testable, so you might need to refactor the needed functionality.

Consider the following "logical" refactoring to what you've presented:

public class MyGui {
    private DataImporter dataImporter;
    public MyGui(DataImporter dataImporter) {
      this.dataImporter = dataImporter;
    }
    public Boolean import(a, b, c) {
       // all UI operations here, 
       // and then when you've gathered all the data:
       byte [] dataToImport = .... 
       return dataImporter.importData(dataToImport, a,b,c);
      
    } 
}

interface DataImporter {
    /*
     * Depending on the configuration parameters a,b,c will import a stream of data identified by byte [] data (again its a schematic example). 
     * Encapsulates logic of data importing based on different parameters
     */
    boolean importData(byte [] data, a, b, c);
}

With this approach, you can test the importing logic without even thinking about the GUI part.

I would try AssertJ framework for Swing GUI testing. They have a repository with the sample projects.

assertj-swing-junit-examples project should be a good start.