The cartoon shows Trump reacting angrily to the missile test, saying "I will not have an unstable egomaniac threaten my country!", as he' s about to thump his fist down on the nuclear button. The joke is that, although Trump is referring to North Korean leader Kim Jong Un, the description could also be applied to Trump himself.
'I will not' is used here with the meaning of 'I'm not prepared or willing to'. One is reminded of Churchill's (possibly apocryphal) comment on the 'rule' that a sentence should not end in a preposition: "This is the kind of arrant pedantry up with which I will not put." See here for more on using 'will'.
An egomaniac is someone who behaves in an unreasonable or crazy way because they think that they and their ideas are extremely important.