Why shouldn't you look at the Sun through a camera using eclipse glasses? One of the NASA approved eclipse glasses manufacturers has the following warning on one of their eclipse glasses:


  Do NOT look through a camera, telescope or binoculars with ANY eclipse shades on!


I can see how the enlarged image of the Sun through a telescope or binoculars might overwhelm the eclipse glasses' filters.

But how would looking at the Sun using a camera's viewfinder or LCD screen reduce the effectiveness of eclipse glasses (assuming you're not using any zoom function)?
Electronic viewfinders (such as the ones using LCD or OLED screens) are excluded from this discussion. There's no danger of eye injury there, except perhaps danger of damaging the camera's sensor if exposure is very long and the lens aperture is big.

It's the classic or optical viewfinders that are targeted by the warning.


  I can see how the enlarged image of the Sun through a telescope or binoculars might overwhelm the eclipse glasses' filters.


A few observations.

It's really the larger aperture of the instrument that is the problem here. The pupil of your eye is only a few mm in diameter. The objective lens or mirror of a telescope could be anywhere between dozens of mm to hundreds of mm (or thousands for very large instruments). The ratio of areas between telescope aperture and eye pupil is even greater - it's the square of the ratio of diameters.

Let's say your eye's pupil is 2 mm in diameter. Let's say you use a 50 mm aperture telescope (small refractor or binoculars). The diameter ratio is 25x. The area ratio is 625x.

All the light captured by the very large area of the instrument is funneled into your eye through your pupil. With the instrument, now you're getting 625x more energy from the Sun, compared to the naked eye view. It's already dangerous to look at the Sun with the naked eye - with the instrument it's 625x more dangerous. And this is with a very small refractor.

There is a class of "solar filters" that are made to be mounted on the instrument's eyepiece, or after the eyepiece. THESE ARE VERY DANGEROUS! All the increased, focused energy of the Sun is now absorbed by the filter, which can warp, melt, crack, or burst into flames. A filter failure at this point is likely to injure the user. White light filtering for solar observations must always occur ahead of the instrument, not after it.



As for why classic, optical viewfinders were included in that list: it was out of an abundance of caution. Most of those viewfinders do not capture more light than your eye does, but some do. It's better to be safe than sorry. You can't expect everyone to be able to tell whether their viewfinder is dangerous or not. So, in a warning addressed to the general population, just tell them to stay away from it.

Now, if you install a full-aperture solar filter (like the Baader solar film) ahead of the viewfinder, then it's safe to use - provided the filter is attached firmly to the camera and cannot be blown away by some random gust of wind.