Egg binding doesn't normally kill the fish, they just get fat and can no longer breed.
She will need soft slightly acidic water and a few males to get her to expel the eggs, she won't do it alone. You have males in the tank so you just need to find out what the GH and pH of the water is. This information can usually be obtained from your water supply company's website (water analysis report) or by telephoning them. If they can't help you, take a glass full of tap water to the local pet shop and get them to test it for you. Write the results down (in numbers) when they do the tests. And ask them what the results are in (eg: ppm, dGH, or something else).
In the wild, neon tetras breed in water with a GH around 0ppm and a pH around 5.5-6.0. In an aquarium they will breed in water with a GH below 50ppm and a pH below 7.0, however, the eggs are unlikely to hatch in this water. But at this stage you just want her to be able to shed any eggs so she doesn't become egg bound, assuming she isn't already.
You can lower the GH by using reverse osmosis or distilled water, or rain water if you can find clean rain water.
You can lower the pH by adding some peat to the water. It will release tannins that stain the water yellow/ brown and lower the pH. The tannins will also encourage the fish to breed.
Doing a big water change each day using slightly cooler water (a couple of degrees) than the tank water can sometimes encourage them to breed. It simulates rainfall and that is when the fish breed, during the wet season.
Raising the temperature a couple of degrees can sometimes help too but water changes and soft acidic water are the main factors to get them breeding.