No, even on rewatch I didn't see that.

But the box isn't mysterious or magical (the mirror motif is everywhere in the Sideways, it's just one more instance).

Here's the explanation: Christian used to visit Claire and her mom while she was little, and he used to sing "to catch a falling star" to her. Claire has mentionned this before (at one point she asked for this song to be sung to her baby as she wouldn't be there to raise him - I think it was in the scene when she was negotiating the adoption of Aaron with the parents at a lawyer's office).

Unlike the original Claire, who met Christian while her mom was dying - and he told her he used to come and sing her songs when she was little - sideways Claire never met him as an adult. That's why she can't connect Christian to the song (at least for now - maybe she'll connect the dots later) or know why he left her a music box that plays the song he used to sing to her.

It also looks like the writers picked a music box specifically as an additional parallel between Claire and Rousseau, who also had a music box playing a song she used to sing (or play) to Alex (can't recall what it played... I remember Rousseau wrote lines from Trenet's "La Mer" on the map that Shannon recognized, but I don't recall if it's the same song the box played). It had been broken for years until Sayid fixed it for her in season 1.