FDA scientists usually have post-graduate degrees, but that's not to say that a college grad is totally out of the running for a job (source). All scientists at the FDA need to have a bachelor's degree in chemistry, biology or some other life science. Research experience in the lab as an undergrad will give you a leg up on other applicants, since the jobs available to those with just a bachelor's degree are typically research positions.
Review scientists are usually more specialized than the basic chemist or biologist, and that can either come from graduate school or years in the workforce. The FDA hires statisticians as well as epidemiologists (they study disease patterns, like how that nasty rash makes its way from a locker room in Minneapolis to a doctor's office in Topeka) and veterinary medical officers. An FDA review scientist may have a PhD in statistics, or might have spent a few years at a pharmaceutical company doing stats. As long as you have plenty of experience, you'll likely be qualified for one of these positions – but the odds are longer if you're coming straight out of college.