Not the case with schizophrenia. It has definite symptoms, as well as chemical signatures (in acute phases).
Schizophrenics have high rates of alcoholism, nicotine addiction, illegal drug use, and homelessness. They are often withdrawn and anti-social, and seldom can hold a job. A very large proportion of homeless people are actually schizophrenic.
They are completely unsuitable to receive gun permits.
Another reasonable cause for disqualification for a gun permit is alcoholism. This is more difficult to assess, because some alcoholics mange (through AA or some other program) to stay clean for long periods of time, or forever. Most, however, rate very unreliable. This might not be dangerous, except that a certain proportion of people become irrationally aggressive when intoxicated.
Most states which issue permits would probably deny them to chronic alcoholics. A drunk-driving record would probably be a good indication that an individual should not get a permit.
Chronic use of illegal drugs should also be a disqualification. This is now a problem, because of the acceptance in many places of the fraud of “medicinal marijuana.”