How did this lead to a uniquely American culture? How did this culture lead us to the "counter-culture" of the 1960s?
When you ask a question like this it would really help if you could tell us what "this" is. I can't be sure of giving a good answer as this question is right now.
People usually say that the counterculture was a response to the culture of conformity that dominated the US during the 1950s. Historians usually say that this culture came around because Amrican adults were so concerned with getting on with their lives now that life was normal. They had lived through the Depression and the War and now they wanted to get ahead. They wanted to make money and have material goods.
Historians also say the conformity came about because of the rise of big companies (so workers needed to fit in to a corporate culture) and because of the McCarthy Era (people didn't want to seem different.
I hope that's what you were asking about.
During the 1950's the American culture was responding to a war in which we had won and they were feeling an air of confidence and peace. The baby boom had occurred and parents were still contained into the conformist world of religious values and American domicile values. Black Americans had been granted the right to vote but still were kept in the place where whites wanted them.
The population suddenly went from the number one group being middle aged to them being teens and young adults. African Americans who had fought alongside the whites during World War II had come home to the same issues they had left behind. The time was right and the energy of the youth helped to spur on the attitudes that it was time for change. Television was now bringing everyday events into the homes of people.
The Vietnam War led to many youths uncertain of why we were fighting. They were afraid and they did not want to fight a war that made no sense to them that was taking so many lives with no outcome in sight. The youths began to rally against the political views of their parents. For the first time the young people had time on their hands. Machines had replaced the chores that they once had to do and many had allowances. They were customers and not working to help provide for their families.
All of the things I mentioned mixed together led to the counter culture that emerged during the 60's.
It should also be noted that the flapper days saw a counter culture emerge.