Has any Sunnah been reported from the Prophet, may Allah bless him and grant him peace, concerning fasting the longest day of the year, which falls on July twenty-first?

There is no evidence in the books of Sunnah (definition) for the recommendation of fasting a specific day in the manner mentioned in the question. Rather, encouragement for fasting has been reported in general, and for specific days linked to the lunar month, such as the White Days, the Day of Arafah, Ashura, and Tasu'a.

Some of the predecessors used to grieve upon death over missing certain acts of worship, among them fasting during intense heat, as narrated from Ibn Umar and Mu'adh ibn Jabal.