For something like this, what I have done myself is create an ArrayList to store the selected cells in. Then refer to that array when you need to know which cells are selected. Then I created a custom cell renderer. In that class, I would check if a cell was in the ArrayList of selected cells, and if it was I would set it to the table cell selected color.
... public class MyTableCellRenderer extends DefaultTableCellRenderer
...
//Defined in your class somewhere
//Add column values to it when clicked on or selected
private final Color selectedColumn = Color.YELLOW;
List<String> selectedCols = new ArrayList<String>();
if (selectedCols.contains(cellValue)) {
tableCell.setBackground(selectedColumn);
} else {
tableCell.setBackground(UIManager.getColor("Table.background"));
}