Please post a link to the page where you are using the shortcode. It usually happens when the player can't find the video. You need to make sure that the video and bucket name are correct.
You can do the following steps which will help you resolve the issue-
First, keep the video public so that you can directly access it from outside. Now create a page and place the following shortcode -
[wp_lightbox_flowplayer_anchor_text_video link="https://s3.amazonaws.com/your-bucket-name/your-video.mp4" width="640" height="480" text="Click here to open the video"]
Does the video play fine in overlay? If it does you can move to the protected shortcode.
Change the video to private again. Now use the following shortcode -
[wp_lightbox_flowplayer_anchor_text_protected_s3_video name="your-video.mp4"
bucket="your-bucket-name" width="640" height="480" text="click here to open the s3
protected video"]
Let me know how it goes.