The Magen Avraham cites a 'practice of individuals' to fast on the Friday prior to the reading of Parshas Chukas (OC 580). In general, it is an anomaly to have a fast day scheduled for a Friday. Of even greater significance is the fact that most fast days are established on a specific calendar date, while this one is not. The Magen Avraham writes that no matter what day of the month the Friday prior to Parshas Chukas falls, that is the day when 'individuals' fast.
What is the significance of this fast day? It commemorates the burning of
20 24 wagonloads of the Talmud and other Sefarim in France. When the event happened, it occurred on the 9th day of Tammuz. However, various Rabbinic authorities of that day learned through dreams that the 'cause' of the incident was not related to the day on the calendar, but to the fact that it was the day before the Torah reading of Parshas Chukas.
The Magen Avraham explains that the Aramaic Targum of the opening words of the parsha [Bamidbar 19:2] "Zos Chukas HaTorah" [This is the law of the Torah] is "da Gezeiras Oraiysa" [this is the Torah's decree]. This was understood to be a Torah decree that such a tragic event would occur on the Friday before this Torah reading.
The Imrei Shammai supplies additional historical background to this incident. He says that in the exact place where the Talmud and other Sefarim were burnt, the Jews of that town had in previous years publicly burnt the Rambam's Sefer - Moreh Nevuchim.
The Moreh Nevuchim was a controversial work. In those days, the Rambam did not yet have the unquestioning allegiance that he gained in later generations. As surprising as it may seem to us, he had his detractors and there were authorities that were highly critical of the Moreh Nevuchim. In fact, there were even some places where his Sefer HaMadah - the first volume of his Major Work "The Yad HaChazakah" was not accepted.
As a Heavenly punishment for this earlier burning of the Rambam's works, 20 cartloads of Torah books were now publicly burnt. When the Jewish community saw this, they recognized their earlier misdeed and repented by establishing a fast day. They prayed for forgiveness and subsequently there was no more controversy about the Moreh Nevuchim.
In this way they were very fortunate. They had a clear Sign from Shomayim in terms of what they had done wrong. It did not take a genius to put two and two together and draw the appropriate conclusion. The connection was obvious. This is the historical background of the custom of 'individuals' to fast on the Erev Shabbos preceding Parshas Chukas.