The user of you program can not citified your Button as a button. Use can think it is a Label. So my Solution is half transparent with appearing & disappearing border.

This code is improved, improved codes, better, better performance, more beautiful, pro-fit.

import java.awt.Color;
import static java.awt.Color.blue;
import java.awt.event.ActionEvent;
import java.awt.event.MouseAdapter;
import java.awt.event.MouseEvent;
import javax.swing.BorderFactory;
import javax.swing.ImageIcon;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JLabel;
import javax.swing.JPanel;

class Test extends JFrame {

    Test() {
        setLayout(null);

        JButton testButton = new JButton("Test_Button");
        testButton.setBounds(200, 200, 200, 20);
        testButton.setOpaque(false); // Must add
        testButton.setContentAreaFilled(false); // No fill
        testButton.setFocusable(false); // I'd like to set focusable false to the button.
        testButton.setBorderPainted(true); // I'd like to enable it.
        testButton.setBorder(null); // Border (No border for now)
        add(testButton);
        
        testButton.addMouseListener(new MouseAdapter(){
            @Override
            public void mouseEntered(MouseEvent e){
                testButton.setBorder(BorderFactory.createLineBorder(blue, 2,true));
                //When enter we can not know our mouse successfully entered to the button. So I'd like to add Border
            }
            @Override
            public void mouseExited(MouseEvent e){
                testButton.setBorder(null);
                //When mouse exits no border.
            }
        });

        // For a half of transparent I'd like a add JPanel behind JButton
        JPanel backPanel = new JPanel();
        backPanel.setBounds(testButton.getBounds()); // Same to buttons bounds.
        backPanel.setBackground(new Color(0, 0, 0, 50)); // Background with transeparent
        add(backPanel);

        JLabel icon = new JLabel();
        icon.setBounds(0, 0, 3840, 2160);
        icon.setIcon(new ImageIcon(getClass().getResource("wallpaper_16105238065ffea49e5ec9d2.04136813.jpeg")));
        add(icon);
        //Background image for test up it.

        testButton.addActionListener((ActionEvent e) -> {
            testButton.setBorder(null); // When mouse clicked border will dissaprear.
            System.out.println("Button clicked!");
            //For check the button works correctly.
        });

        setSize(800, 500);
        setVisible(true);
    }

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

My output