JTable Cell Renderer

Add this line:

c.setOpaque(true);

The Component returned by getTableCellRendererComponent must be opaque in order to see changes on background and foreground color. The problem here is also another: you are extending DefaultTableCellRenderer (that is a JComponent) but you are returning a Component that hasn't setOpaque method. I would refactor your code like this:

public Component getTableCellRendererComponent(JTable table, Object value, boolean isSelected, boolean hasFocus,int row,int col) {

    String s = table.getModel().getValueAt(row, col).toString();
    this.setOpaque(true);
    if (s.equalsIgnoreCase("yellow")) {
        this.setForeground(Color.YELLOW);
    }
    else {
        this.setBackground(Color.WHITE);
    }

    return this;
}

Is your renderer ever even used? You make it the default renderer for cells containing String, but have you overloaded your model's getColumnClass method so that it knows that some of the cells hold Strings?

So first I'd use println statements to see if the renderer is even being called and if not, I'd override my model's method as noted above.

Edit 1
Also your if results are bound to be strange. In the if portion you change the forground and in the else you change the background -- makes no sense. You probably should do complementary changes in state in the if vs. the else blocks, not orthogonal changes.

Edit 2
For example:

import java.awt.*;
import java.util.Random;

import javax.swing.*;
import javax.swing.table.*;

public class Board extends JPanel {

   private static final long serialVersionUID = 1L;

   int boardHeight = 20;
   int boardWidth = 10;

   JTable table;
   Random random = new Random();

   public Board() {
      setLayout(new BorderLayout()); // !!
      DefaultTableModel model = new DefaultTableModel(boardHeight, boardWidth) {
         @Override
         public Class<?> getColumnClass(int columnIndex) {
            return String.class;
         }
      };
      // !! table = new JTable(this.boardHeight, this.boardWidth);
      table = new JTable(model);
      for (int row = 0; row < model.getRowCount(); row++) {
         for (int col = 0; col < model.getColumnCount(); col++) {
            String s = random.nextBoolean() ? "red" : "yellow";
            model.setValueAt(s, row, col);
         }
      }
      table.setDefaultRenderer(String.class, new BoardTableCellRenderer());

      table.setFocusable(false);
      table.setShowGrid(false);
      table.setRowMargin(0);
      table.setIntercellSpacing(new Dimension(0, 0));
      table.setRowSelectionAllowed(false);
      table.setVisible(true);
      this.add(table);
      this.setPreferredSize(new Dimension(table.getPreferredSize().width,
               (table.getPreferredSize().height + 85)));
   }

   private static void createAndShowUI() {
      JFrame frame = new JFrame("Board");
      frame.getContentPane().add(new Board());
      frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
      frame.pack();
      frame.setLocationRelativeTo(null);
      frame.setVisible(true);
   }

   public static void main(String[] args) {
      java.awt.EventQueue.invokeLater(new Runnable() {
         public void run() {
            createAndShowUI();
         }
      });
   }
}

class BoardTableCellRenderer extends DefaultTableCellRenderer {

   private static final long serialVersionUID = 1L;

   public Component getTableCellRendererComponent(JTable table, Object value,
            boolean isSelected, boolean hasFocus, int row, int col) {

      Component c = super.getTableCellRendererComponent(table, value,
               isSelected, hasFocus, row, col);
      Object valueAt = table.getModel().getValueAt(row, col);
      String s = "";
      if (valueAt != null) {
         s = valueAt.toString();
      }

      if (s.equalsIgnoreCase("yellow")) {
         c.setForeground(Color.YELLOW);
         c.setBackground(Color.gray);
      } else {
         c.setForeground(Color.black);
         c.setBackground(Color.WHITE);
      }

      return c;
   }
}

Here is a simple solution, use TableCellRenderer as an inner class.

    myTable.setDefaultRenderer(Object.class, new TableCellRenderer()
    {
        JLabel comp = new JLabel();
        String val;

        @Override
        public Component getTableCellRendererComponent(
                             JTable table, 
                             Object value, 
                             boolean isSelected, 
                             boolean hasFocus, 
                             int row, 
                             int column)
        {
            comp.setOpaque(true);
            comp.setForeground(Color.BLACK); // text color

            if (value != null)
            {
                val = value.toString();
                comp.setText(val);

                if (val.equalsIgnoreCase("red"))
                {
                    comp.setBackground(Color.RED);
                }
                else if (val.equalsIgnoreCase("yellow"))
                {
                    comp.setBackground(Color.YELLOW);
                }
                else if (val.equalsIgnoreCase("green"))
                {
                    comp.setBackground(Color.GREEN);
                }
                else
                {
                    comp.setBackground(Color.WHITE);
                }
            }
            return comp;
        }
    });