See e.g. "Light-Induced Damage to the Retina" http://photobiology.info/Rozanowska.html
With a little more thought maybe all these warnings exaggerating the dangers of the sun to your eyes are a good idea. It seems that humans know or learn "Don't stare into the sun" is a good idea but during an eclipse some people forget that.
But how do you know they didn't get eye damage? Seems reasonable to think that people did sustain damage to varying degrees. It still happens today... not even that rare, actually.
Also, assuming they weren't running for cover due to mistaking a total solar eclipse as a bad omen, they may have been interested in other things that prevented them from staring for extended periods. The 360 degree sunset, for example.
I'd imagine the people of the time would think this was some sort of negative sign from their sun god (Horus) and madly prostrate or panic. I think it'd be abnormal for people to stare it as they'd be too afraid but that's just conjecture.
Note that sea captains in the age of the sextant were typically blind in one eye by using the sextant on the one star visible during the day (our sun). But this wasn't from a single viewing, it was from many many viewings, presumably the majority of which were while they were midshipman. They were smart enough to sacrifice one eye, covering the other while looking at the sun.
Nevertheless it's a bad idea, especially with both eyes, and eclipses are more dangerous.
Doesn't it hurt when you look at the sun? Any time I've ever accidentally glanced at the sun it's literally a blinding pain and my eyes just close reflexively.
It has never made sense to me when people say, "Don't stare at the sun". It's roughly the same as saying, "Don't stick your hand in the fire." Well duh, it hurts.
Used to do that during long car trips to pass the time, I don't have any issues with my sight as an adult but it's definitely a necessary warning.
How does your civilization know NOT to watch an eclipse with a naked eye?