in java the operand to test equality between two items is == not '=' which is an assignment; an assignment returns the assigned value, so your :
if (playerOne = true)
will always be true as playerOne will be assigned to true, then the if will become if (true) and the statement associated will always be executed.
the best way to refactor your code is:
public void mouseClicked(MouseEvent arg0) {
if(playerOne) {
playerOne = false;
playerTwo = true;
boxOne.setIcon(xIcon);
} else if(playerTwo) {
playerOne = true;
playerTwo = false;
boxOne.setIcon(oIcon);
}
}
as the something == true would be redundant.