Before we examine those claims, it's important to ensure readers have a proper general understanding of the Bill of Rights. Contrary to common misperception, these amendments do not bestow privileges upon American citizens. Rather, they are primarily a set of prohibitions against the government infringing on pre-existing human rights all people have.
That's evident in the language. For example, the First Amendment begins "Congress shall make no law…" This amendment isn't awarding citizens the rights of religion, speech and assembly—it's outlawing the government's thwarting of those innate and universal human rights.
Similarly, the Fourth Amendment asserts that "the right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated." Again, the authors are not granting those rights, they are protecting them.
When the Bill of Rights was proposed, some feared the enumeration of a handful of rights could be misinterpreted as providing a comprehensive catalogue—and thus empowering the government to infringe on human rights not specified. That's why they included the Ninth Amendment, asserting that "the enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people."
"Amendment II. A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed."