I'm not sure how accurate this is, but my current perception of the Vietnam War is essentially that of the US seeing the Vietnamese people democratically choosing a communist government and intervening in their affairs solely due to fear of spreading communism. This led to a war that was completely unsupported by the Vietnamese outside of the extreme minority and officials in Diem's government, which was essentially a puppet dictatorship -- that is, conflict in Vietnam began as an anti-imperialist war against the French, after which the country would have peacefully established an independent government that would have happened to be communist, but the US took over for the French and began a brutal aggressive war for the sake of "saving the Vietnamese from themselves." This position is mostly based on
(a): Eisenhower stating that 80% or more of the Vietnamese public would have supported a communist government, and
(b): The fact that Diem's puppet state was brutally authoritarian and repressive, and maintained its power only because of consistent US support.
Due to this view, am I being unfair in characterizing the South Vietnamese government and military essentially in the same bin as Vichy France in WWII? Clearly the Vichy regime did include officials and soldiers who were French, but the vast, vast, vas majority of the population was awaiting liberation from the occupying Nazis, and those who worked with the Vichy government and army were merely collaborationists and essentially traitors working with an occupying power -- wouldn't the same apply to the South Vietnamese, who were essentially fighting on the side of an invading imperialist power against their own nation's right to self-determination?
Your current perception is a popular one, but unfortunately based on older scholarship that is now wholly out of date. To accurately answer your question, one has to first and foremost consider historiography.
In the early American historiography of the Vietnam War, whether or not the author felt that the war was justified or not, there were large generalizations about North and South Vietnam. The tendency to draw the entire population of what was essentially two nations into one easily identifiable category of simple peasants who wanted their freedom for their nation was misguided and was the result of authors who could not read Vietnamese, who didn't know really know these countries as intimately as a scholarly expert or who simply focused on what was more important: The American perspective.
Thankfully, the scholarship of the Vietnam War has begun to move away from such easy generalizations and now puts a focus on aspects that have either been ignored or simply overlooked. The most important historiographical change was the so-called "Vietnamese Turn" in which a larger focus has been given to South and North Vietnamese topics to explore the impact of the war in every sector imaginable of both nation's societies. In the last ten years alone, there has been an explosion in fantastic scholarship into South Vietnam. This has meant that familiar South Vietnamese figures (who have been turned into stereotypes) like Ngô Đình Diệm have been reassessed and given a new voice. But most importantly, the orthodox view of South Vietnam as either a puppet state or a pawn has been made obsolete. South Vietnam is now portrayed as its own actor in the Vietnam War era, alongside North Vietnam and the United States (as well as the myriad of other nations involved).
This brings us to your question: "To what degree did those in South Vietnam outside of the government support the Vietnam War and the establishment of a non-communist state?"
The answer to that, as any other historian would say about the population of any region/nation/kingdom/empire (etc.) is that it's more complicated than that. There were no simple two sides. In South Vietnam, people made their own choices and that had tremendous impact on their lives and that of their families. There were those who actively protested against the South Vietnamese government (who in turn violently suppressed them) but who had no love for communism or North Vietnam. There were others who hated the South Vietnamese government because of their policies (such as conscription) or who were the victim of the Army of the Republic of Vietnam's heavy hand. Political diversity existed both in the cities and on the countryside, although rural South Vietnam can be a difficult subject to tackle due to the lack of primary sources. Furthermore, historians such as Edward Miller has produced scholarship surrounding Ngô Đình Diệm that gives us a more nuanced understanding of the man and his government, showing that he was not some strange and foreign puppet put into place by the United States, and that South Vietnam was not a passive national, adhering strictly to the will of the United States. Instead, nationalism in South Vietnam is traced to the early 20th century to show the plurality of nationalist thought and how that continued into the 1960s.
Speaking for myself as a researcher, my own research into the Kit Carson Scouts who were former PAVN/PLAF soldiers who defected to the South Vietnamese government and then chose to work for the United States show that political diversity in action on both sides. Some were men who had been forcibly conscripted by the PLAF and who took the first chance to escape while others were men who felt disillusioned by North Vietnam, politically or militarily, and chose to defect rather than dying. Between these two examples, we find a large amount of different explanations or motivations but it all comes down to one simple fact: They made their own choices. It was not obvious to them that they were "fighting for Vietnam" or that "they were never going to surrender" -- indeed, they did surrender and chose to throw in their lot with the opposite side.
Needless to say, this is a complicated topic and one that is difficult to summarize. Asking what an entire nation of millions felt about a particular subject is a very difficult if not impossible question to answer. Yet what modern scholarship has done is to balance that previous image of South Vietnam and brought up new, fresh perspectives. The outdated image of the corrupt South Vietnam will live on but perhaps there will be a new, popular re-imagining that will bring some nuance to the popular view of the war's participants.
Suggested reading:
South Vietnamese Soldiers: Memories of the Vietnam War and After by Nathalie Huynh Chau Nguyen (Praeger Publishing, 2016).
Vietnam's Forgotten Army: Heroism and Betrayal in the ARVN by Andrew Wiest (NY Press, 2009).
ARVN: Life and Death in the South Vietnamese Army by Robert K. Brigham (University Press of Kansas, 2006).
Cauldron of Resistance: Ngo Dinh Diem, The United States, and 1950s Southern Vietnam by Jessica Chapman (Cornell University Press, 2013).
Misalliance: Ngo Dinh Diem, the United States, and the Fate of South Vietnam by Edward Miller (Harvard University Press, 2013).
Hanoi's War: An International History of the War for Peace in Vietnam by Lien-Hang T. Nguyen (University of North Carolina Press, 2012).
Van Nguyen-Marshall, "Student Activism in Time of War Youth in the Republic of Vietnam, 1960s–1970s," Journal of Vietnamese Studies 10, No. 2 (Spring 2015): 43-81.
Nu-Anh Tran, "South Vietnamese Identity, American Intervention and the Newspaper Chính Luận [Political Discussion], 1965-1969." Journal of Vietnamese Studies 1, no. 1-2 (February/August 2006): 169-209.
Edward Miller, "The Postcolonial War: Hue-Tam Ho Tai and the “Vietnamese Turn” In Vietnam War Studies," Journal of Vietnamese Studies 12, no. 3 (2017): 14-22.
Sean Fear, "The Ambiguous Legacy of Ngô Đình Diệm in South Vietnam’s Second Republic (1967-1975)," Journal of Vietnamese Studies 11, no. 1 (2016): 1-75.