Can American Citizens Lose Their Citizenship?

Yes, but only if they intend to relinquish it (or, if they are naturalized citizens and committed fraud during the naturalization process).

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. The relevant federal statute, 8 U.S. Code § 1481, provides (emphasis added):

A person who is a national of the United States whether by birth or naturalization, shall lose his nationality by voluntarily performing any of the following acts with the intention of relinquishing United States nationality—

(1) obtaining naturalization in a foreign state [as an adult];

(2) … making … [a] formal declaration of allegiance to a foreign state [as an adult];

(3) … serving in … the armed forces of a foreign state if (A) such armed forces are engaged in hostilities against the United States, or (B) such persons serve as a commissioned or non-commissioned officer; or

(4) (A) … serving in … any … employment under the government of a foreign state [as an adult] if he has or acquires the nationality of such foreign state; or

(B) … serving in … any … employment under the government of a foreign state [as an adult] … for which … employment … [a] declaration of allegiance is required; or

(5) making a formal renunciation of nationality before a diplomatic … officer of the United States in a foreign state …;

(6) making in the United States a formal written renunciation of nationality … whenever the United States shall be in a state of war and the Attorney General shall approve such renunciation as not contrary to the interests of national defense; or

(7) [being convicted of] committing any act of treason against, or attempting by force to overthrow, or bearing arms against, the United States, violating or conspiring to violate any of the provisions of section 2383 of title 18 [rebellion or insurrection], or willfully performing any act in violation of section 2385 of title 18 [advocating overthrow of government], or violating section 2384 of title 18 [seditious conspiracy] by engaging in a conspiracy to overthrow … by force the Government of the United States, or to levy war against them ….

But as both the statute and Afroyim make clear, the conduct listed in (1) through (7) will only surrender the person's citizenship if the person engaged in it "with the intention of relinquishing" citizenship. Subsection (b) of this statute provides that "Any person who commits or performs, or who has committed or performed, any act of expatriation under the provisions of this chapter or any other Act shall be presumed to have done so voluntarily, but such presumption may be rebutted upon a showing, by a preponderance of the evidence, that the act or acts committed or performed were not done voluntarily." That presumption is relevant when a person claims that one of the expatriating acts was done under duress, e.g., that he was forced to serve in an enemy army. But

It is important … to note the scope of the statutory presumption. [The subsection] provides that any of the statutory expatriating acts, if proved, are presumed to have been committed voluntarily. It does not also direct a presumption that the act has been performed with the intent to relinquish United States citizenship. That matter remains the burden of the party claiming expatriation to prove by a preponderance of the evidence….

This post today was triggered, of course, by President Trump's saying he is "giving serious consideration" to revoking Rosie O'Donnell's citizenship. I have no reason to think that O'Donnell has engaged in any of these acts, such as (in her case) "going through the process of obtaining her Irish citizenship through her grandparents," "with the intention of relinquishing United States nationality." American law allows dual citizenship, and "does not require a U.S. citizen to choose between U.S. citizenship and another (foreign) nationality. A U.S. citizen may naturalize in a foreign state without any risk to their U.S. citizenship."