Firstly just double check that you're using the correct package names as you've got the package listed as EXTENSION_PACKAGE but then add application.receivers to the definition in your manifest. It probably should read like the following:
<uses-permission android:name="android.permission.RECEIVE_BOOT_COMPLETED" />
<application>
<receiver android:enabled="true" android:name="EXTENSION_PACKAGE.ApplicationStartupReceiver" android:permission="android.permission.RECEIVE_BOOT_COMPLETED">
<intent-filter>
<action android:name="android.intent.action.BOOT_COMPLETED" />
<category android:name="android.intent.category.DEFAULT" />
</intent-filter>
</receiver>
</application>
Now when the receiver is called I'm assuming you want to simply start your AIR application? To do this is a little more complex than just creating an Intent with the actions you are specifying. Android hasn't got way of handling the actions you are calling. What you need to do is find your application entry from the context and start an intent with that, as below:
if (Intent.ACTION_BOOT_COMPLETED.equals(intent.getAction()))
{
String componentName = context.getPackageName() + "/.AppEntry";
Intent i = new Intent( Intent.ACTION_MAIN );
i.setComponent(ComponentName.unflattenFromString( componentName ));
i.addCategory( Intent.CATEGORY_LAUNCHER );
i.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
context.startActivity( i );
}