As it was pointed out in the comments, you probably don't need both arrays. When both combo boxes should contain the same values, then you can pass the same array to both combo boxes.
The actual question seems to be aiming at how to perform a specific action depending on the combination of the selections of two combo boxes.
I think there are two options for this: You could either store the combo boxes as instance variables, or you could store the selections of the combo boxes as instance variables. Since you're already storing the combo boxes as instance variables, this should be the easier one to go here. So you could do something like this:
@Override
public void actionPerformed(ActionEvent e)
{
String color1 = (String)combo1.getSelectedItem();
String color2 = (String)combo2.getSelectedItem();
// Possibly check if either color is 'null' here
if (color1.equals("blue") && color2.equals("yellow"))
{
message.setText("green");
}
...
}
(Note: If you now intend to write a nested switch-statement like
switch(colour1)
{
case "red":
switch(colour2)
{
// 10 cases...
}
break;
// 10 x 10 cases...
}
you should think about a different approach, depending on what you want to do with these colors...)