No, you couldn't strip flag-burners of citizenship, even if flag-burning could be made a crime

Nobody should be allowed to burn the American flag - if they do, there must be consequences - perhaps loss of citizenship or year in jail!

- Donald J. Trump (@realDonaldTrump) November 29, 2016

Contrary to President-elect Donald Trump's tweet, even if flag-burning weren't protected by the First Amendment (and it is), you couldn't strip people of their citizenship for it.

Let's begin with the constitutional text, here from section 1 of the 14th Amendment:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.

Once you have American citizenship, you have a constitutional entitlement to it. If you like your American citizenship, you can keep your American citizenship - and that's with the Supreme Court's guarantee, see Afroyim v. Rusk (1967):

There is no indication in these words of a fleeting citizenship, good at the moment it is acquired but subject to destruction by the Government at any time. Rather the Amendment can most reasonably be read as defining a citizenship which a citizen keeps unless he voluntarily relinquishes it. Once acquired, this Fourteenth Amendment citizenship was not to be shifted, canceled, or diluted at the will of the Federal Government, the States, or any other governmental unit.

(Special bonus in Afroyim: a cameo appearance by a Representative Van Trump in 1868, who said, among other things, "To enforce expatriation or exile against a citizen without his consent is not a power anywhere belonging to this Government. No conservative-minded statesman, no intelligent legislator, no sound lawyer has ever maintained any such power in any branch of the Government.") In Vance v. Terrazas (1980), all the justices agreed with this principle.

Now, as with almost all things in law - and in life - there are some twists. Naturalized citizens can lose their citizenship if they procured their citizenship by lying on their citizenship applications; the premise there is that legal rights have traditionally been voided by fraud in procuring those rights. And citizens can voluntarily surrender their citizenship, just as people can generally waive many of their legal rights; this surrender can sometimes be inferred from conduct (such as voluntary service in an enemy nation's army), if the government can show that the conduct was engaged in with the intent to surrender citizenship.

But flag-burning, whether or not it is intended to express contempt for the United States (and burning an American flag, like flying the Confederate flag, can have many possible intentions), is generally not accompanied by an intent to renounce U.S. citizenship, nor is it generally evidence of any such intent. A college student's expression of contempt for the college's administration, or the college as a whole, doesn't mean an intent to drop out of the college - it's entirely consistent with an intent to make the best of a bad situation, or even to take advantage of the benefits provided by an institution that one despises. One might consider such an attitude dishonorable, depending on the circumstances, but it's very plausible that the contemptuous student would have that attitude. That is even more clearly so as to a citizen's expression of contempt for the current American administration, or even America as a whole (if that's the flag-burner's attitude), given how costly surrender of citizenship would be, especially when one lacks another country that will take one in.

So even if flag-burning could be made criminal (and, I note again, it can't be), the 14th Amendment protects the flag-burner's citizenship, just as it protects other criminals' citizenship.