If you are "looking for an example of a scientist saying "this is my evidence for it"", then the answer is no, there is no credible evidence of it. If there were, it would be common knowledge because it would be so out of the ordinary.
Also, it's interesting to note that for the Sun to stop moving across the sky, that would actually mean the Earth would have (almost) stopped spinning, because it's the Earth spinning that causes the Sun to appear to move.
For fun, if the Earth stopped spinning suddenly...
The ground under your feet, and your body itself, is currently moving at about 1000 miles per hour (less if you are closer to the poles) due to the Earth spinning. If the Earth suddenly stopped spinning, you would be thrown to the East at that speed, like when you hit the breaks hard in your car at 60 mph you are thrown at the windshield. The oceans lakes and streams would be thrown as well, waves of water would crash over everything.
Also, the Earth is not a perfect sphere, it's actually wider than it is tall, due to the force generated by the spinning motion (think of spinning on your feet -- your arms will be pulled outwards). This also includes the water on the Earth. So, if it were just the Earth ceasing to spin, then the oceans would (after stopping their 1000 mph scouring the landscape), due to the force of gravity, start settling to create a spherical shape -- which would flood the north and south of the Earth and expose the ocean floors around the equator!
The atmosphere is also spinning. Suddenly there would be 1000 mph winds from west to east. Not that you would feel them -- you'd be flying inside them.
Of course, when stopping the Earth from spinning, it would be a good idea to stop all the things on and in it from spinning, too!
Anyway, it's all illustrative of the fact those verses are written by/for people who don't know that it is the Earth that moves around the Sun and not the other way around.
I'm not sure the Earth or Sun stopping for a day that long ago in history could be proven scientifically other than by it being recorded by people. So a related question to ask would be, did anyone else record the Sun stopping at that time in history? This should have been a world wide event, unless it was some kind of local illusion. What other civilizations existed at the time, and did any of them record it? However, it's hard to find similar evidence for things we know happened -- for example, Halley's Comet appears every 75-76 years, but until only about 1000 years ago, it is hard to find multiple accounts of it from the same year. It may have been recorded twice in 467 BC when a comet was recorded both in Greece and in China (check the Halley's Comet Wikipedia page).