Short answer:

The Spanish flu wasn't nearly so deadly as the plague; a case fatality rate of perhaps %2.5 vs %60 or more for plague (approaching %100 for septicaemic or pneumonic plague). Prisons were unusual before the modern era, so we're really only talking about the 1918-19 flu having an impact on a large incarcerated prison population.

Discussion:

There were not substantial prison populations at the time of the Justinianic Plague or Black Death of the 14th century. Long term incarceration of large numbers of people is a relatively new phenomenon.

It could be said that quarantine was itself "a prison" -- those with or suspected of having the plague were "shut up" in their own homes. Daniel Defoe famously compares the quarantined to prisoners, and we can empathize.

But beyond the metaphorical while people might be sent to the galleys or sold as a slave, typically offenses weren't often punished by prison-- prison is expensive, and the growth of residential incarceration comes from the 19th century on.

Convicts on the galleys do have an interesting role in Early modern European plague; France has galleys through the 18th century, up to the Revolution-- we have accounts that galley slaves l were used to bury the dead in Marseilles, and they themselves contracted plague in large numbers:

The pestilence was caused by the wickedness of the people and the indignation of Providence ; it was cured by the intelligence and activity of the local officials. There were first the sheriffs, "than whom the Roman consuls, so full of the love of their country, did certainly never carry their zeal to so high a pitch ; " then there was M. Lebret, the first President of the Parliament, who exerted himself (in absentia) "with such goodness and condescension;" the officers in command of the galleys, "not one of whom to save the city would not have hazarded his own life," but who satisfied themselves more cheaply, by sacrificing over 250 of the galley slaves, who were drafted into the infected city and forced to bury the dead, a service which proved fatal to the most of them.

So the 1918-1919 flu was the only one of the pandemics which occurred at a time with a substantial prison population.

Prison populations were easy to study, and the course of influenza in San Quentin prison was written up by the resident physician. He identified the likely causes of new outbreaks -- arrival of prisoners-- and also had a good clue as to the incubation period, due to the Sunday night movies where large numbers of inmates mixed, which were followed some days later by new peaks in cases.

Assuming that this respiratory infection attacked its victims at the Shows on Sunday, it would seem that there was an incubation period of from 36 to 60 hours, which produced the sudden illness on the following Tuesday or Wednesday. A typical history of many of the cases is that on Sunday they visited the show, and that on Tuesday or early Wednesday they were seized with headache, fever, chills, bone ache, severe prostration, and sometimes nausea

Dr Stanley also tried equipping the prisoners with masks, but he doubted that it did any good. They also developed a quarantine system for those who became ill, and an isolation system for new arrivals to the prison.

There's a particularly politically significant connection with Irish prisoners -- there are reports of 100 Sinn Fein prisoners ill in Belfast jail -- curiously none of them died, which made them much healthier than other places. Perhaps the stringent security, restricting access to other people may have protected them . . . though in Usk prison there are fatalities, allegations of poor medical care and political pressure to release prisoners.

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Sources:

Beiner, Guy, et al. “Greatest Killer of the Twentieth Century: The Great Flu of 1918-19.” History Ireland, vol. 17, no. 2, 2009, pp. 40–43.

“The Plague Of Marseilles.” The British Medical Journal, vol. 2, no. 1502, 1889, pp. 827–829.

Stanley, L. L. “Influenza at San Quentin Prison, California.” Public Health Reports (1896-1970), vol. 34, no. 19, 1919, pp. 996–1008.