Store them in an airtight container, and when you share them, you can put them in Ziploc bags decorated for the occasion.
If you want to be certain what type of worm it is (and this can help you know how to treat the worms), wipe the worm off your cat's behind with a piece of tissue, and put it into a Ziploc bag with a small piece of damp paper towel.
You should also take a stool sample in a clean container such as a Ziploc bag.
While you can put your child's lunch items into Ziploc baggies, you can cut down on costs by purchasing some small Tupperware or Gladware containers.
Most dogs are ready to have a bowel movement first thin in the morning, so you'll want to have a clean Ziploc bag ready and label it with a marker.