This code worked for me while providing additional information for the intent. This is not video data, but image data, but I believe that the concept should be the same (I could be wrong, since I did not try it on video).
File file = new File(getExternalFilesDir(null), "image.png");
String uriPath = "file://"+file.getPath();
shareIntent.putExtra(Intent.EXTRA_STREAM, Uri.parse(uriPath));
, , , , ( getExternalFilesDir (null)). .
, , , - , ( mp4 ?)