Dear Azizi,
Yes, you are correct in assuming that the word "BOSCA" meaning "box" does come from the Irish language (sometimes incorrectly called "Gaelic" when in fact the correct term is "Gaeilge", pronounced gwail-geh)
I'm inclined to agree with you that the similarities for words meaning "woods", such as the Portuguese and Spanish "bosque", the Dutch "bos", the French "bois", and the Italian "bosco" (would you believe!!) might well tie in with the fact that until quite recent times most boxes were made of wood.
I grew up in Ireland and never encountered either the American product, or the associated jingle, and I have been living in Cornwall, UK, for the last 15 years and I have not come across them here either.
Best regards,
Anne XX