First off, the answer is no. There are no laws prohibiting private employers from requiring personality tests before offering you a job. This is so even though personality testing is often an inaccurate means of determining your abilities in the job, and even though the test takers can answer the questions in any way they choose to trick the exam. Unfortunately, these personality tests can often hold biases against members of a minority class, though if this is the case, and it can be proven, the test would then be against the law.
Furthermore, if a personality test asks questions meant to elicit religious beliefs, sexual practices, or race norms, the test is illegal, and you can sue over a test that is deemed to be psychologically invasive, though such lawsuits are very expensive and difficult to win.