Before we dwell deeper into American war crimes during the Vietnam War, it is necessary to say that this is still a highly debated topic within the studies of the American involvement in Vietnam. Without a shadow of a doubt, we know that there were war crimes such as the infamous My Lai massacre in 1968.
The question that many historians, and you, have asked is just how extensive were the killing of civilians?
One of the biggest challenges for a historian dealing with American war crimes during the Vietnam War is how to confirm an alleged war crime. There were many veterans that returned from their tour in Vietnam, spoke openly about seeing some truly horrific events or even killing civilians themselves. One such case are the infamous Winter Soldiers who all appeared on camera talking about their experiences. The Army's Criminal Investigative Division, whose responsibility it is to investigate alleged atrocities in the war, considered 46 out of 109 testimonies to warrant an investigation. Out of these 46, only one individual was found to have sufficient evidence to continue the investigation which in turn confirmed it. The official CID reports on the other 45 individuals state that a good deal of them refused to answer any questions or cooperate (for different reasons, personal and not) while some men changed their initial testimonies. See this thread for more information on the difficulties involved with confirming atrocities during the Vietnam War.
Returning to the original questions: How extensive were American war crimes in the Vietnam War?
Looking on the extreme side of alleged war crimes, we can get numbers such as 3000 civilians alone between 1 December 1968 to 1 April 1969 in IV Corps Tactical Zone. These atrocities were allegedly purported by the 9th US Infantry Division. That's clearly a massive number for such a short period of time, especially considering that the My Lai massacre (and My Khe, which was also a massacre that occurred simultaneously as My Lai, but did not receive the same national attention) in total had around 400 to 500 killed. I would rather point, however, to confirmed cases to give us a clearer view: The Son Thang massacre where 16 civilians, 11 of whom were children, were killed by Marines from the 1st Marine Division as well as 30 civilians that were killed by soldiers from the Americal division during Operation Iron Mountain. These numbers are far more common. The killing of civilians in small numbers were widespread. The My Lai and My Khe massacres were the exception rather than the rule, but historians such as Bernd Greiner make the case for smaller but far more common killings of civilians.
The question the reader might be asking right now is why atrocities like these would happen? Why would American soldiers kill South Vietnamese civilians? I've written about this here.
Now, to the question of free fire zones:
Free fire zones has been used by many nations involved with counterinsurgency, if not always by that name. It is the supposed consequence of resettlement in which the populace is either forced or encouraged to move from their present location to a location where they are more easily controlled.
This too happened during the Vietnam War: The populace living in proposed fire free zones were either encouraged or forced to relocate. When that had been done, everyone else remaining inside were deemed hostile at the creation of the fire free zone. The most infamous of these free fire zones was the "Iron Triangle" north of Saigon which was known as a strong VC area. During Operation Cedar Falls, the ARVN and South Vietnamese police moved in to gather the villagers, screen them (sometimes with forceful interrogations) and then resettle them, commonly by force.
What is true is that as soon as the area was deemed free of civilians, it would indeed be allowed to treat anyone as hostile which in turn obviously led to atrocities and war crimes.