Hickox makes two arguments for not needing quarantine. First, while in New Jersey, she twice tested negative for Ebola. However, this is inconclusive since, as this article from Vanity Fair indicates:
When Ebola strikes, it kills quickly, but it can take up to three weeks to incubate, and usually around 10 days. The period is long enough that contact with a possible source may have been forgotten, and long enough for infected people to travel without symptoms. And even if you tested for Ebola—which nobody in Guinea had the capacity to do—you wouldn’t find it during the incubation period: Ebola can’t be detected in the blood until symptoms show. An epidemic can start slowly and go unnoticed for weeks. ...(Underline mine). In other words, until she becomes symptomatic, it may not be possible to detect the Ebola; but she may not become symptomatic for up to 21 days. That is why she needs to stay in quarantine.
Her second argument is that she cannot, as she alleges, spread Ebola through simple contact. "'You could hug me, you could shake my hand [and] I would not give you Ebola,' she said." Unless she coughed or sneezed into her hand, or had bodily fluids get on her clothes. How does she think colds and the flu spread?