I have animal-specific eye wipes at home, but they're not always handy. Am I putting my dog in danger of an eye infection or other ailment if I wipe the corner of the eyes with my fingers?
I do all the time! That being said, if you notice any odd colored eye gunk, or that the eye is constantly running, you should probably see a vet for possible infection. I think the "regular eye boogies" are just that, and it's OK to swipe them away with a clean hand (fingers) as long as you are swiping away, not across the eye area.
Wipes or damp towel are best, as they said, but in a pinch your clean fingers will work... watch out if you've got longer nails and be sure there are no open wounds (hangnails, etc) to transfer fluids.
Yes you can use your hands however make sure your hands are clean so no bacteria gets into their eyes. Wipes are typically best or a wet paper towel!
I use my hands all the time, but make sure you wash them first.
You really shouldn't use your hands, the eye area is very delicate. Best to use a cotton ball dipped in warm water to clean the eye area. There are also wipes made for cleaning dogs eyes. Keep the hair near the eyes short that's helps with keeping it clean. There is also small combs made for the eye area.