ETA: I just did some research and you're right, sometimes the eggs don't show up on a fecal test. It's important to test anyway, just to see if any other worms are present. If you tell them there are little white segments in her poop than they'll probably give you a wormer that kills tapeworms just in case. It's not harmful to her or anything.
Yep, sounds like tapeworms. I don't know where you read that a fecal won't show tapeworms but it most definitely should? At least ours did...
Franklin had tapeworms as a puppy. It's no big deal at all. I know it seems gross but they just give him a dewormer (either a pill or liquid) and depending on the medicine he'll need either one or multiple doses. They are quite common and super easy to treat.
She'll be fine.