What is the ruling on someone who broke their fast on Saturday, having fasted on Friday, because they remembered the prohibition against singling out Friday for fasting? Is it permissible for them to form the intention to fast on Saturday during the day and complete that day's fast?

It is not a condition to form the intention the night before for voluntary fasting. A person may intend to observe a voluntary fast after Fajr, provided they have not committed any acts that invalidate the fast. Scholars are unanimous in deeming it disliked to single out Friday for fasting, unless it coincides with a fast one customarily observes, or if one fasts the day before or the day after it.