What is the ruling on hanging a paper in the house that includes the family's expectations for the gender of the newborn (male or female)?

The unseen is known only to Allah, alone. Expecting certain outcomes based on experience or repetition of an event, or anticipating what has already shown evidence, is not considered claiming knowledge of the unseen. Rather, it can fall under firasah (sagacity or intuitive insight). What the questioner's family did by mentioning their expectations is merely conjecture and speculation, which is permissible, Allah willing. It is narrated regarding firasah that the gender of the fetus can be known, as happened with Abu Bakr al-Siddiq, may Allah be pleased with him, in his expectation that his wife's pregnancy would be a girl. And conjecture (dhann) can be obligatory, recommended, forbidden, or permissible. Obligatory is having a good opinion of Allah, and forbidden is having a bad opinion of Allah and of Muslims, while permissible is like the expectation of Abu Bakr.