1. Will worms effect eggs? Should I not eat the eggs?
2. Will worms cause chicken to stop laying? She was laying and has stopped. I'm not sure if it's due to the move to our home or the worm.
3. How do I treat?
Here's link with information about worms.
Some worms can affect eggs but most don't. If you cook your eggs thoroughly they will die. It can stop laying because it causes stress, they absorb nutrients needed to produce the egg and can drain the energy out of them to lay an egg. There are many ways. I've heard of some people using Oregano oil to worm them and stabilize their birds, there are some wormers. Some say mix in water I prefer to inject orally that way I know that they are getting the worming agent. With worming it's best to do treatment for 5 days, wait 2 weeks and then treat for another 5 days. I've also used foods like garlic and onions that have sulfur in them.
Welcome to BYC. Those look like round worms, which are the most common kind in chickens. Other worms such as cecal, and capillary worms, are harder to see. Valbazen sheep and cattle wormer 1/2 ml orally to each chicken, and repeated in 10 days will get most worms in chickens. Then make sure that your coop has clean and dry litter that you stir often.
Welcome to BYC!
The OP needs to use a proper wormer, DE will not help.
Diatom or DE is pretty ineffective when wet inside the GI tract, so it is not an effective wormer. It also is not that good to breathe in. It's probably a better approach to deworm on a twice a year or more schedule with a proven product such as Valbazen or fenbendazole (SafeGuard, Panacur.)