Chicken egg with "sand" in it another with it on the shell--read and I will explain

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Ok so my girls like to lay wherever, usually in the coop but I have to hunt. One day I got an egg i suspected had been buried in the shavings from the day before so I marked it and put it in the fridge. i didnt want to sell this one if it was old, those become dog food. Anyway I cracked it open a couple days later to find tiny little white things that I thought might be some sort of eggs. eww! Or mold from being in the coop maybe longer than the day and a half? LOL I had no idea but i threw that one out. Ok so the next day or two I went out and got an egg and it had the same thing on the OUTSIDE. These were tiny little things that again, look like eggs from a parasite maybe? Pure white, perfectly round and when I wiped them off (they were not an attached part of the shell like extra calcium, although they did resemble little calcium sand particles) they were like sand and just cme right off. This grossed me out since it was the second time I had seen them. It seems maybe I have also had a couple of shellless eggs as i have found yolk messes but no shell to be seen anywhere, not even a small bit. I got my hens from some people who were starting a fresh young flock, these are 2-3 years old. They said they were raised on organic food, but I have to say, the hens are less than gorgeous, some eggs had shells so thin they were hard to pick up without cracking and oddly shaped or with rough calcium deposits. I feed them well on regular laying crumbles/pellets and they free-range and get table scraps when available but seeing the thin shells i got oyster shell for them as well and the eggs seem to be getting much more "normal" looking and harder shells. My hen, the only one I had left after a mishap with a FOX, always lays nice hard eggs, hard to crack open, without the aid of oyster shell in her diet. I will ask even though I dont want to hear the yucky answer but were these possible worm eggs of some sort? Seeing how they were inside the egg, then outside on the shell. I am SO grossed out to think if they do have worms, and i sell an egg with worms in it....oh my goodness. Help!!!