There is a 20% chance of a hidden BGA appearing when loading the song from Music Select. With that being said, you cannot hit Restart and expect the video to change. You must go back to Music Select and load the song to try again. You must clear it to unlock it in the Collection. For the achievements related to hidden BGAs you only have to see the hidden BGA for it to unlock. You do not need to clear the song for the achievements.
.U-NIVUS has the hidden BGA. The regular BGA shows a black background with LENA appearing afterwords. The hidden BGA shows a storyboard background with a drawing of LENA.