Allow a Tab key press in a TextArea to focus on the next field and an Enter key press in a TextArea to trigger a default button.

TextAreaTabAndEnterHandler.java

import javafx.application.Application;
import static javafx.application.Application.launch;
import javafx.beans.value.*;
import javafx.collections.ObservableList;
import javafx.event.*;
import javafx.scene.*;
import javafx.scene.control.*;
import static javafx.scene.input.KeyCode.ENTER;
import static javafx.scene.input.KeyCode.TAB;
import javafx.scene.input.KeyEvent;
import javafx.scene.layout.VBox;
import javafx.stage.*;

public class TextAreaTabAndEnterHandler extends Application {
  final Label status = new Label();

  public static void main(String[] args) { launch(args); }

  @Override public void start(final Stage stage) {
    final TextArea textArea1 = new TabAndEnterIgnoringTextArea();
    final TextArea textArea2 = new TabAndEnterIgnoringTextArea();

    final Button defaultButton = new Button("OK");
    defaultButton.setDefaultButton(true);
    defaultButton.setOnAction(new EventHandler<ActionEvent>() {
      @Override public void handle(ActionEvent event) {
        status.setText("Default Button Pressed");
      }
    });

    textArea1.textProperty().addListener(new ClearStatusListener());
    textArea2.textProperty().addListener(new ClearStatusListener());

    VBox layout = new VBox(10);
    layout.setStyle("-fx-background-color: cornsilk; -fx-padding: 10px;");
    layout.getChildren().setAll(
      textArea1, 
      textArea2, 
      defaultButton, 
      status
    );

    stage.setScene(
      new Scene(layout)
    );
    stage.show();
  }

  class ClearStatusListener implements ChangeListener<String> {
    @Override public void changed(ObservableValue<? extends String> observable, String oldValue, String newValue) {
      status.setText("");
    }
  }

  class TabAndEnterIgnoringTextArea extends TextArea {
    final TextArea myTextArea = this;

    TabAndEnterIgnoringTextArea() {
      addEventFilter(KeyEvent.KEY_PRESSED, new TabAndEnterHandler());
    }

    class TabAndEnterHandler implements EventHandler<KeyEvent> {
      private KeyEvent recodedEvent;

      @Override public void handle(KeyEvent event) {
        if (recodedEvent != null) {
          recodedEvent = null;
          return;
        }

        Parent parent = myTextArea.getParent();
        if (parent != null) {
          switch (event.getCode()) {
            case ENTER:
              if (event.isControlDown()) {
                recodedEvent = recodeWithoutControlDown(event);
                myTextArea.fireEvent(recodedEvent);
              } else {
                Event parentEvent = event.copyFor(parent, parent);
                myTextArea.getParent().fireEvent(parentEvent);
              }  
              event.consume();
              break;

            case TAB:
              if (event.isControlDown()) {
                recodedEvent = recodeWithoutControlDown(event);
                myTextArea.fireEvent(recodedEvent);
              } else {
                ObservableList<Node> children = parent.getChildrenUnmodifiable();
                int idx = children.indexOf(myTextArea);
                if (idx >= 0) {
                  for (int i = idx + 1; i < children.size(); i++) {
                    if (children.get(i).isFocusTraversable()) {
                      children.get(i).requestFocus();
                      break;
                    }
                  }
                  for (int i = 0; i < idx; i++) {
                    if (children.get(i).isFocusTraversable()) {
                      children.get(i).requestFocus();
                      break;
                    }
                  }
                }
              }  
              event.consume();
              break;
          }
        }  
      }

      private KeyEvent recodeWithoutControlDown(KeyEvent event) {
        return new KeyEvent(
          event.getEventType(), 
          event.getCharacter(), 
          event.getText(), 
          event.getCode(), 
          event.isShiftDown(), 
          false, 
          event.isAltDown(), 
          event.isMetaDown()
        );
      }
    }
  }
}

Solution to StackOverflow question: JavaFX 2 TextArea: How to stop it from consuming [Enter] and [Tab]

Thanks for sharing your code. It inspired me to improve on your work.
Your solution doesn't work in most scenarios. For example, I have a TextArea and a Label inside a HBox, and your code will only search for nodes within that HBox, it does not attempt to look for siblings of the HBox.
I have created my own Gist which searches the Node graph in a more complete manner.

https://gist.github.com/nickthecoder/57ac20829214f1209e41e851d955ed35

My Gist concentrates only on the "focus next" part, not on the keyboard event filters in you solution.

Sorry, it is written in Kotlin, not Java, but I think it will be easy to port to Java.