The first reason the answer to this question is NO! is an obvious one, ethically, it's a big no-no to diagnose someone you've never met before. You wouldn't tell your client that his mom has Borderline Personality Disorder, for example. The only people we can analyze, diagnose and recommend treatment for are our clients. Diagnosing requires meeting with an individual and often hearing private information they don't disclose to others.
The second reason is that an ethics rule the APA created in direct response to a politician taking legal action against Psychiatrists who called him unfit for the presidency because of certain mental illnesses. In 1964, Barry Goldwater successfully sued the Psychiatrists quoted in the article for libel, and the APA gave us "The Goldwater Rule", which states:
On occasion psychiatrists are asked for an opinion about an individual who is in the light of public attention or who has disclosed information about himself/herself through public media. In such circumstances, a psychiatrist may share with the public his or her expertise about psychiatric issues in general. However, it is unethical for a psychiatrist to offer a professional opinion unless he or she has conducted an examination and has been granted proper authorization for such a statement.
Sure, we all have personal, private thoughts about each of the candidates. It may be tempting to offer a professional opinion about someone based on the behaviors they exhibit, but proceed with caution.