The free software movement arose out of a belief that software ought to be free and available for anyone to use and modify. Did the counterculture movement and Free Love have anything to with that? Also, most software companies were established by people who were born during the baby boom and who would have come of age during the height of counterculture, the 70's. Would it be right to say that the software industry, and by extension Microsoft, Apple and the first wave of technology companies (I'm not talking about IBM and Xerox) and much of what it accomplished was influenced by counter culture?
Open source software and the free sharing of software were both integral components of software development prior to the counterculture movement of the 1960s.
Take a look into the SHARE movement. The group was founded in 1955 by users of the IBM 701 for the purpose of sharing software. IBM released their OS and users took it upon themselves to develop and share software. Users benefited from better software without incurring additional costs (an important fact considering computers were being leased for thousands of dollars) and IBM benefited from the free labor of individuals dedicated to improving the functionality of their products.
I could do some more research and give you a long and drawn-out answer of the history of software but I think I've already answered your question.
If you feel like getting a little philosophical, I think the answer to your question is mostly irrelevant. The concepts of free sharing and open source have existed long before software or the counterculture movement. Any developer who claims to have adopted these concepts from the 1960s counterculture movement is ignoring the fact that they have existed since the birth of humanity and dominate our lives on a daily basis. For example, every recipe you've ever made is the result of a long history of free sharing and open source "code".