The court document challenging the imprisonment of these two chimpanzees, Leo and Hercules, filed by The Nonhuman Rights Project, has the words "writ of habeas corpus" crossed out. See the document here. While the judge, Barbara Jaffe, originally did not strike out these words in the first document, she has since amended it.
|Portion of the amended court document|
In December of 2014, a court in Argentina appeared to give an orangutan, named Sandra, human rights, but once the court document was properly translated into English, it became clear the media originally erred and this was not the case. Whether or not Sandra is entitled to human rights is now a confusing matter, but this no longer looks like the landmark case it was made out to be.
As of today, it appears that apes have not been granted the same rights as humans in the United States. They have not been declared non-human persons and have never been granted the writ of habeas corpus.
Links of interest:
How human are chimps?
The Nonhuman Rights Project
Jane Goodall's stance on chimpanzees and their rights