Why do people fry their eyes staring at the sun during a solar eclipse? Is it that they stare at the sun for a long time---longer than if they stared without an eclipse? Or is it that during the eclipse the sun's light rays somehow become more dangerous?
I know that the Moon's gravitational pull will affect light (it's basically the principle that allows gravitational lensing), but I doubt the effect even amounts to one angstrom, hardly enough to change their nature.
People generally avoid staring at the sun; it's bright and painful and will damage your eyes. During an eclipse people want to watch the eclipse happening and without approved darkened glasses they will damage their eyes. The light is just as bright and just as damaging to your retina. The only time you can view the eclipse safely without safety glasses is when it is total, then you are not viewing the sun itself only its corona.
see this link
The first rule of enjoying the eclipse is to avoid looking directly at the sun without eye protection. Even brief glances can cause permanent damage.
The only exception to this rule is for lucky spectators in the path of totality during the few minutes of the total eclipse, when the sun is fully blocked by the moon.
That link will also tell you three ways to safely view the eclipse (indirectly) without glasses.
As for gravitational lensing, you are right that the moon's effect is immeasurable. When Eddington confirmed in 1919 Einstein's GR the effect of the sun was to deflect light by only 1.75 arcseconds link
During the darkest parts of the eclipse, your pupils may dilate to adjust to the darkness. A reasonable hand-waving estimate for the difference in area between a fully-constricted pupil versus a fully-dilated pupil is a factor of 100. This means that looking directly at a 95% eclipsed Sun might put one twentieth of the Sun's light onto your face — but if your pupil is exposing fifty times more area than usual, you might actually get more sunlight on your retina than staring directly at the full sun. The eclipsed Sun might also fail to trigger your startle/squint/lookaway reflex.
