The release of all prisoners of war taken by all sides, parallel to the withdrawal of all US troops from Vietnam, was agreed by the US and the North Vietnamese as part of the 1973 Paris Peace Accords (Chapter III, Article 15), and this part of the agreement appears to have been honored as the release and withdrawal took place within the first few weeks of the accord being signed, long before the cease-fire called for in the agreement had deteriorated enough for South Vietnamese leader Nguyen Van Thieu to declare a de facto resumption of hostilities in January of the following year.

Accounting for all Americans who were killed or captured in the war however has proved to be not just emotionally fraught but also politically contentious, as theories have been floated by some sectors of the American public accusing the communist Vietnamese of holding on to secret prisoners after the war's end (which, as you mention, provided the premise for the plot of the 1985 action blockbuster Rambo: First Blood Part II), but these allegations have never been appreciably substantiated, apart from the controversial claims of Robert Garwood PFC (a "stay-behind" who was court-martialed upon his return to the US in 1979 for collaborating with the communists between his 1969 capture and 1973 release (link to the DoD's report)).

The United States and Vietnamese governments have both denied these allegations, and those who believe in them typically accuse US leaders of intentionally covering up the truth to avoid embarrassment. A Select Committee on POW/MIA Affairs was convened to investigate the matter, but its 1993 findings conclude that there is "no compelling evidence that proves that any American remains alive in captivity in Southeast Asia." The report details the committee's process in determining the likelihood of any number of the 2,264 "unaccounted for" Americans surviving after 1973 as secret captives and its reasons for regarding this is as improbable, and is worth reading in its entirety if you want a fuller understanding of the issue.

Unsurprisingly, the United States has never sponsored any search-and-rescue missions for alleged captives whom its leaders have consistently maintained do not exist. However, a series of private trips to Vietnam and Laos, allegedly in search of these abandoned POWs, were undertaken in the early 1980s by Lieutenant Colonel James Gordon "Bo" Gritz, a Vietnam veteran-turned-conspiracy-theorist, whose trips were heavily publicized despite failing to locate any of the alleged POWS and probably directly inspired the plot of Rambo II. Gritz's activities won him the support of Texas billionaire Ross Perot and later that of neo-Nazi publisher and Holocaust-denier Willis Carto, who convinced him to run for president in 1988 and 1992 on the Populist Party ticket. Gritz became a prominent leader in the far-right Patriot/militia movement of the 1990s as a member of the white supremacist and anti-semitic "Christian Identity" network after appearing in support of Aryan Nations militant Randy Weaver during the 1992 Ruby Ridge standoff. (link to Bo Gritz's SPLC extremist file)

The role of the nationalist "Abandoned Vietnam POWs" narrative in catalyzing Gritz's descent into fascism, I think, invites comparison to the "Stab-In-The-Back Legend" popular among German veterans of WWI and its role in the rise of the original Nazi movement.