Don’t forget: Flash != Flex. Flex uses elements from Flash but is built on own classes. If Flash and Flex should play together we need a interface.
MovieClip is a display object in Flash and UIComponent is a display object in Flex. But we can wrap a MovieClip into a UIComponent, so the Flex UIComponent becomes a container for our Flash MovieClip.
Here both files, at first the MXML file:

And the Actionscript file with the class:

package {
	import mx.core.Application;
	import mx.core.UIComponent;
	import flash.display.MovieClip;

	/*****************************************************************
	 * File: FlexApp.as
	 *****************************************************************/

	public class FlexApp {
		private var _app:Application;

		public function FlexApp(app:Application) {
			_app=app;

			var btn:MovieClip=new BtnStop();
			var flexui:UIComponent=new UIComponent;
			flexui.addChild(btn);
			app.addChild(flexui);
		}

		static public function initApp():FlexApp {
			return new FlexApp(Application(Application.application));
		}

		public function get app():Application {
			return _app;
		}
	}
}

How it works

applicationComplete in the MXML file is the eventhandler called from the application. Here we provide our handler, which must be a static member function of a class so it can be called later.
initApp() creates our class and passes a application object to the constructor.
Inside the constructor we create the MovieClip from the imported Flash symbol “BtnStop” (this symbol comes from a swc lib, created with Flash). Then we create a Flex UIComponent container and add our MovieClip as a child. The UIComponent can now be added as child of the Flex application.

Overview of development tools for Flash/Flex

My recommendation:

For debugging, visual design of MXML: FlexBuilder
For mostly coding of Actionscript under Windows: FlashDevelop.