http://graphics-geek.blogspot.com/2010/05/video-flex-4-path-to-enlightenment.html
Using that information, in ActionScript, I added a Group and then added my component to the group.
Now I can detect whether my component was clicked and I can even change the stroke! I borrowed the hit test code from Chet's Top Drawer. Thanks to Chet and Guillaume!
//MXML
<s:Panel id="panel" width="563" height="373" x="76">
<s:layout>
<s:TileLayout>
</s:TileLayout>
</s:layout>
<s:Panel>
//ActionScript in MXML
//This is a bit klugey, but it works.
var mySymbol:MyComponent = new MyComponent();
var group:Group = new Group();
var i:IVisualElement = panel.addElement(group as IVisualElement);
group.addElement(mySymbol);
mySymbol.addListeners();
//ActionScript in component
//in the addListeners function:
this.parent.addEventListener(MouseEvent.MOUSE_DOWN, handleMouseDown);
//in handleMouseDown
var selectPoint:Point = localToGlobal(new Point(event.localX, event.localY));
//parent is the Group that was added in the MXML code
if (this.parent.hitTestPoint(selectPoint.x, selectPoint.y))
{
this.stroke = new SolidColorStroke(0x000000,2);
}