New shelter cat has white worms on poop. Will my other cats get them?

It sounds like Eli might have tapeworms - these usually look like small rice granules and can be seen around the anus or on top of the stool. Tapeworms are usually spread through fleas and cannot go directly from one cat to another, so if your other cats are on flea preventative there may not be a concern. Roundworms are longer and look more like spaghetti inside the stool and can be transmitted through stool from cat to cat. I would recommend keeping Eli separated from the other cats with a separate litter box until you can see a veterinarian - you can bring a stool sample with you so they can determine what type of worms are present and prescribe medication to treat them.

It's also fairly common for cats who were dewormed at the shelter to still have some intestinal parasites - it can take several doses of dewormer and not all deworming medications cover all parasites.