Cutworms? Put a small "collar" of aluminum foil around the base of each seedling and see if the damage stops.

I'm having problems with cucumber seedlings this year too. None of the transplants I've put out, just Middle Eastern cucumbers germinating in hills.
They aren't cut to the ground. Mine are the leaves being picked off and in most cases, disappearing?
My garden is fenced with 1/2" wire mesh, so I can't see it being rabbits. Squirrels or birds I think.
Perhaps I'll follow your lead and put chickenwire cages around the hills and see if it solves the problem.

Well I tried the foil but a couple of them were still eaten off. I've also put out moth balls, hung a foil pie pan, and put a small fence around the entire garden. None of which has helped.I'm considering putting clear plastic bottles around individual plants to see if that helps. Only issue I can see there is drainage when it rains. I'm thinking some small holes in the bottom might help with that.