I had the same problem this year. I told DD how to make the eggs & I think she turned them down before the water came to a boil.
I was able to peel them w/o losing too much & then chopped them for egg salad. I had to use my pastry cutter to do it. I wasn't thrilled with it, but at least it was edible. We usually eat the egg salad in pita, so having it be extra sloppy wasn't too bad.
The only other idea I have would be to chop them as best you can & add them to soup. I've never re-boiled the eggs once I refrigerated them so I don't know if that would work - or even if it's safe. I did it once when they were still warm & that worked. (DD was 3 & couldn't wait for a hard-boiled egg so I peeled one as soon as I took it out of the pot. I had to toss it though - so much for trying to avoid the screaming.

)