No, it isn't true.
The unflattering portrait of Donald Trump is not banned in the USA. Satire is protected in the U.S.. This portrait, or reproductions of it, are fully protected free speech. What is not protected, is profiting off the likeness of an individual. No federal law protects the Right to Publicity, but some states do.
Donald Trump could sue in New York State, because NY specifically protects people's likeness from being used unauthorized in commercial works. It specifically has an exception for fine art, which this portrait might qualify as. If the artist wasn't trying to sell the painting, or if they released it to the public domain as in Dastar Corp v. Twentieth Century Fox, Trump would have no civil recourse.
Donald Trump's campaign website doesn't have an official position on the Freedom of Speech or the First Amendment. He has made several public statements though. In an interview (4:38 mark), he disagreed with SCOTUS's Citizens United v. FEC decision.
"I guess from my standpoint personally I'd almost rather not see it," he said of candidates seeking to raise large amounts of money from the Koch brothers. "I see all of the money that's being raised by these folks, and they're raising hundreds of millions of dollars, and ultimately billions of dollars."
Trump has made the self-funded nature of his campaign a signature issue, insisting that unlike his rivals he cannot be bribed as he won't be in thrall to donors. "I don't need anybody's money," he said, but "I'm not sure it's the worst thing in the world."
He has also spoken publicly about strengthening the libel laws so the press isn't protected from publishing lies.
"One of the things I'm going to do if I win, and I hope we do and we're certainly leading. I'm going to open up our libel laws so when they write purposely negative and horrible and false articles, we can sue them and win lots of money."
The press is currently protected when telling lies, unless a plaintiff can prove they acted with malice, in reckless disregard to the truth (New York Times Co. V. Sullivan).