Things I would fix:
-------------------

Don't do this:

    import javax.swing.*;
    import java.awt.*;

It could be problematic for the compiler to import a bunch of packages at once. If two packages provide the same type, both are imported, and the type is used in the class, a compile-time error occurs. This is described in [JLS 6.5.5.1][1]:

> Otherwise, if a type of that name is declared by more than one type-import-on-demand declaration of the compilation unit, then the name is ambiguous as a type name; a compile-time error occurs.

In addition, it also saves a bit of memory.  And your IDE (if you use one), should have the ability to do this automatically.

---

[The serializable class `DrawPanel` does not declare a `static final serialVersionUID` field of type `long`][2] (thanks to @GilbertLeBlanc for the link):

    private static final long serialVersionUID = 1L;

---

Don't **ever** do this:

    catch (Exception exc)
    {
    }

Say that you do get an [`Exception`][3] thrown at you. You aren't handling it right now in any way.  At least print a stack trace to help you solve problems you may encounter in the future:

    catch (Exception exc)
    {
        exc.printStackTrace();
    }

It was also mentioned in the comments to catch specific `Exceptions`.  Here there is no real need to do that since only an [`InterruptedException`][4] is being thrown.  When you do need to catch multiple `Exceptions`, [some people might tell you to do this][5]:

    catch (IOException io)
    {
        io.printStackTrace()
    }
    catch (InterruptedException ie)
    {
        ie.printStackTrace();
    }

Recommendations that are optional:
----------------------------------

You have a lot of `if` statements with braces:

    if(down)
    {
        oneY++;
    }

If you don't expand on them (which I don't see a need to), you can save some LOC:

    if (down) oneY++;

---

Right now you are setting the location:

    frame.setLocation(375, 55);

You don't have to do this, but I prefer the [OS to set the location][6]:

    frame.setLocationByPlatform(true);

You could also set the [frame to the center of the window][7]:

    frame.setLocationRelativeTo(null);

But I never liked that too much.  It makes your program seem like a pop-up, which I despise.

---

I always do `frame.setVisible(true)` after I am completely finished setting up the frame.

---

Right now your are doing this with your braces:

    void someMethod(int someInt){
        // ...
    }

Since you are a beginner, I would recommend lining up your braces (I still do this, and I've been programming for a while):

    void someMethod(int someInt)
    {
        // ...
    }

---

You might note that this:

    public static void main(String... args)

Is a bit different than your usual:

    public static void main(String[] args)

They both do the same thing, but one uses [variable arity parameters][8].

Final code:
-----------

    import java.awt.BorderLayout;
    import java.awt.Color;
    import java.awt.Graphics;
    
    import javax.swing.JFrame;
    import javax.swing.JPanel;
    
    final public class Test
    {
    
    	JFrame frame;
    	DrawPanel drawPanel;
    
    	private int oneX = 7;
    	private int oneY = 7;
    
    	boolean up = false;
    	boolean down = true;
    	boolean left = false;
    	boolean right = true;
    
    	public static void main(String... args)
    	{
    		new Test().go();
    	}
    
    	private void go()
    	{
    		frame = new JFrame("Test");
    		frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
    
    		drawPanel = new DrawPanel();
    
    		frame.getContentPane().add(BorderLayout.CENTER, drawPanel);
    
    		frame.setResizable(false);
    		frame.setSize(300, 300);
    		frame.setLocationByPlatform(true);
    		frame.setVisible(true);
    		moveIt();
    	}
    
    	class DrawPanel extends JPanel
    	{
    		private static final long serialVersionUID = 1L;
    
    		public void paintComponent(Graphics g)
    		{
    			g.setColor(Color.BLUE);
    			g.fillRect(0, 0, this.getWidth(), this.getHeight());
    			g.setColor(Color.RED);
    			g.fillRect(3, 3, this.getWidth() - 6, this.getHeight() - 6);
    			g.setColor(Color.WHITE);
    			g.fillRect(6, 6, this.getWidth() - 12, this.getHeight() - 12);
    			g.setColor(Color.BLACK);
    			g.fillRect(oneX, oneY, 6, 6);
    		}
    	}
    
    	private void moveIt()
    	{
    		while (true)
    		{
    			if (oneX >= 283)
    			{
    				right = false;
    				left = true;
    			}
    			if (oneX <= 7)
    			{
    				right = true;
    				left = false;
    			}
    			if (oneY >= 259)
    			{
    				up = true;
    				down = false;
    			}
    			if (oneY <= 7)
    			{
    				up = false;
    				down = true;
    			}
    			if (up) oneY--;
    			if (down) oneY++;
    			if (left) oneX--;
    			if (right) oneX++;
    			try
    			{
    				Thread.sleep(10);
    			}
    			catch (Exception e)
    			{
    				e.printStackTrace();
    			}
    			frame.repaint();
    		}
    	}
    }


  [1]: http://java.sun.com/docs/books/jls/second_edition/html/names.doc.html#32799
  [2]: http://stackoverflow.com/questions/285793/what-is-a-serialversionuid-and-why-should-i-use-it
  [3]: http://docs.oracle.com/javase/7/docs/api/java/lang/Exception.html
  [4]: http://docs.oracle.com/javase/7/docs/api/java/lang/InterruptedException.html
  [5]: http://stackoverflow.com/questions/21938/is-it-really-that-bad-to-catch-a-general-exception
  [6]: http://docs.oracle.com/javase/7/docs/api/java/awt/Window.html#setLocationByPlatform%28boolean%29
  [7]: http://docs.oracle.com/javase/7/docs/api/java/awt/Window.html#setLocationRelativeTo%28java.awt.Component%29
  [8]: http://stackoverflow.com/questions/2925153/can-i-pass-an-array-as-arguments-to-a-method-with-variable-arguments-in-java/2926653#2926653