This is because despite being a Veteran with 2 Purple Hearts Snake believes he was betrayed by the United States government
Some time later, he turned to a life of crime due to the perceived betrayal of the United States government during the "Leningrad Ruse" (when he lost the use of his left eye) and when his parents were burned alive in their home by the United States Police Force—events described in the Escape from New York novelization by Mike McQuay. He traveled with his war buddy and only friend, Bill Taylor. Snake took up with partners Harold Hellman (later known as "Brain") and Fresno Bob. In Kansas City around 1993, Hellman apparently let Plissken and Fresno Bob get cornered by police, at which time Fresno Bob was brutally tortured and killed by sadistic law enforcers within the United States Police Force.
Source: Snake Plissken > Blackground (2nd Paragraph)
his dislike for the government is shown again at the end of the film after the President only offers half-hearted regret for the people who died saving the president
As the President prepares for a televised speech to the leaders at the summit meeting, he thanks Snake for saving him and tells him that he can have anything he wants. All Snake asks for is how he feels about the people who died saving him, but the President only offers half-hearted regret. As Snake walks away in disgust, Hauk offers Snake a job as his Deputy- Snake demures, neither saying yes or no. The President's speech commences, and he offers the contents of the cassette; to his embarrassment, the tape is Cabbie's cassette of the swing song "Bandstand Boogie". As Snake walks away, he intentionally tears the magnetic tape, out of the cassette reel, with the actual message that was intended to be delivered by the President.
Source: Escape from New York > Plot (Last Paragraph)
infact Snake (if i recall) is parroting the Presidents own words in the quote in the question, so it's really rich of the President to be saying that to Snake
but really Snake didn't give a crap about what happens to the US or to the president by that point. everything he did in New York he did because he had been injected with Microexplosives and Snake is a survivor
This of course repeats in Escape from LA where Snake is infected with a virus to kill him and is forced to go into former LA to recover the "Sword of Damocles", a super weapon being a series of satellites capable of rendering all electronic devices anywhere on the planet useless, which was stolen by the President's Daughter Utopia, in exchange for a full pardon and a cure to the virus.
At the start of the film the President already tells Snake that he doesn't care if Utopia is returned considering her a traitor and by the end of the film it's revealed that
the Virus Snake was infected with was nothing for that a fast acting strain of the flu (which is revealed after the time limit they let expire) and the President orders Utopia to be taken to an electric chair to be executed
the Prescient then tries to use the Sword of Damocles but fails to wipe out the Cuban invasion force, revealing that what they actually took was a "Map to the Stars" Guide. after the President orders Snake to be executed but fails (because Snake activated a hologram and escaped) Snake proceeds to activate the Sword of Damocles himself but using the World Code which wipes out all electronics worldwide.
So in Escape from LA not only did the president lie to Snake and show no concern for Utopia but they didn't trust Snake either, a decision that had Snake didn't do anything the US would have been invaded by Cuba.