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Small white worms tend to be segments of tapeworms. They are usually contracted by ingesting fleas. They are gross because they can crawl out of the rectum. Dried tapeworms look like small rice pieces.
I would medicate both dogs. Friday there should be a vet open to pick up the pills, you should be able to pick it up without a visit.
For now I would wipe the hindend keep the dogs down and wash your hands. You are not to worry about spreading, just the good hygiene.
Good Luck, and let me know if you have further concerns.
No, it is not the same. Tapeworms are their own little type of worm and they need their own wormer. The most common is called DRONCIT and is about $4/pill. You need one for now and I would repeat it in 2-3 weeks.