I can tell you Unico was an anime movie series by Osamu Tezuka (who did AstroBoy) and that despite what it says on the site, was never a television series. The "intro" that is uploaded is the opening sequence to one of the 2 dubbed Unico movies released in the US.
There were 2 movies. "The Fantastic Adventures of Unico" and "Unico and the Island of Magic". I only remembered parts of the first movie. But I bought a homemade DVD of it and I was able to remember more parts. I saw the squirrels that scared my brother, lol. I always thought that his mom and siblings were killed, but they were only frozen for awhile while Unico was taken away. It was the first anime I ever saw. I know there's a website somewhere on the net if you google it that you may find and see more pictures.
Wow. This is a really obscure memory for me. All I remember is the white little unicorn with pink hair, and a point in one of the movies where a goddess flies through the sky, blanketing it into night with her cloak. It the first anime for me, too. I remember feeling strange and disconcerted at certain moments when I was small.