You can worm them with a few things. Wazine will take care of round worms only, but round worms tend to be the most common worm. You can get this at most feed stores. 1 ounce to one gallon of water for one full day. Repeat in 12 days. If you suspect other types of worms, you can use Safeguard horse paste wormer. You can get this at feed stores as well. A pea sized dolup on one tiny piece of bread. Then repeat in 12 days. Either one of these will help to worm your birds. If you are getting worms in your eggs, the worms are crawling up from the intestinal tract and heading down the oviduct and getting into the eggs before the shell is put on.
I thought worms ending up in eggs was very rare. The OP has not even ascertained what the things are in the egg. I have never wormed my chickens & don't understand why every question asked turns into the need to worm the chickens. I understand it is sometimes necessary, but not every problem a chicken has goes back to having worms.
I agree, the description does not sound like worms. Most likely blood or meat spots, especially if they can be ruptured when you put pressure on them. any parasitic worm I've ever seen is pretty rough, and would not rupture easily.
Worms are in the intestinal tract. The egg gets its shell before it's close enough for that. Worms would not receive nutrition if they lived inside the oviduct.
OP, if you can please post a picture. I suspect meat spots but it's difficult to tell without seeing a picture.