It is rare to see them in the eggs; they are different body cavities so something abnormal has to happen. They can even have a heavy infestation without seeing them in the poop, but that would be the more normal place. Many vets will do a fecal smear on a sample for a reasonable fee, although a negative fecal doesn't mean there are no worms. Many people worm them routinely, though others don't at all.
Chemical wormers intended for goats, cows, etc.. are the only way to kill all worms. You have to toss the eggs for a few weeks. There are natural wormers that reduce the load but will not kill them all. It's just something you have to read up on, then make your own choice.
If you post the chemical and brand name of the product you bought, we may be able to tell you whether people use it on chickens.