Let me state that at the outset that I don't agree with Donald Trump's proposal that Japan and South Korea should develop their own nuclear weapons given our existing security arrangements with the two countries.
White House Deputy National Security Adviser Ben Rhodes stated, "It would be catastrophic were the United States to shift its position and indicate that we support somehow the proliferation of nuclear weapons to additional countries."
Yet given the eagerness and enthusiasm with which the Obama Administration entered into a nuclear agreement with Iran, Rhodes' statement is lacking in credibility. Should Saudi Arabia and the other Gulf Arab states develop nuclear weapons then the Iran nuclear deal will have served as its primary impetus.
Even if it less than ideal for Japan and South Korea to develop nuclear weapons if given the choice of countries I'd like to see to develop such systems which countries would you rather see enrich uranium? I would suggest that most people would place Japan and South Korea much higher than Iran.