SSD and nand chip life is reduced by the writing to it, not the reading of it.
Hashing files stored on the SSD will be reading them, even hashing it over and over again would be reading them, so a SSD would be very good for that type of operation. Even with writing, you would have to write massive quantities before that would be a real issue in modern SSD drive.
The quality of the drive itself could be just as important as how it was treated. A SSD would be much faster for the operation, depending on other factors, like CPU speed, Bus speed for communicating to the drives, and even the efficiency and optimisation of the program doing the hashing.
You did not cover in your question if you will be copying the files to the drive, or repeatedly copying files to there to just do the hash, then remove them again. That (of course) would not be the same.
A Hard disk could just as easily do this task for years, and writing to it over and over again, makes little difference, the hard drive would be much slower for the disk reads and writes.
Either one of them or any operations your doing that involve storage , would still benefit from having a proper backup of the data, preferably disconnected from the computer after created, where it can sit safely if anything (soft or hard) would go wrong with either the SSD or a HD.