I've heard this a lot, but as far as I am aware the number of hippies wasn't actually very high, so I imagine some fudging is done about just people from the 60's voting republican, as opposed to hippies in particular. But a cursory googling isn't finding me much good stuff about this.
No, for the most part, they did not.
There is little truth to the idea that as one grows older one also grows more conservative. In actuality, age cohorts (people who were born around the same time and had similar cultural experiences) gravitate towards one end of the ideological spectrum and then tend to stay there for the rest of their lives. Individuals will, of course, often change ideologically, but age cohorts do not do this.
When we talk about the political movements of the 60s and what those people are doing in today's politics, we have to keep three things in mind:
The Silent Majority. The silent majority is the term for the people who were not at Woodstock or out protesting but were doing what most people do most of the time - getting by, having a job, starting families, etc. The "hippies" pull all of our attention but they were not in the majority.
The issues have changed. The major issues of those days were, in many ways, different from the major issues of today. I can't, of course, give you a platform that every "hippie" would have agreed upon but some general principles were an end to the war in Vietnam and a more open and tolerant culture. There very well may have been a large number of "hippies" then who also wanted lower taxes and less government involvement in business transactions who would feel right at home in the modern GOP.
What it means to be a Republican has changed. The Republican Party underwent a big shift under Ronald Reagan, in part, as a response to the "hippies" of the 1960s and the "Me Generation" of the 1970s. Ronald Reagan used a very social conservative and hawkish tone that switched a lot of non-liberal Democrats over and those people, especially in the South, still vote Republican today. Let me be clear on one point here: it is not that Reagan changed these peoples' ideology - rather it is that he had a message that appealed to already existing political feelings. And by welcoming these people into the party and making his message the new tone of the party he changed a lot about what it meant to be a Republican.
The best piece of work I have seen on this entire question is here:
Which, itself, is pulling information from this paper:
http://graphics8.nytimes.com/newsgraphics/2014/07/06/generations2/assets/cohort_voting_20140707.pdf
Looking at this information we can see that people adopt an ideology based upon life experiences in their teens and young adulthood then remain there for their entire lives. We can see that the current generation of people coming of age will, in contrast to the baby boomer generation, likely stay moderately liberal for the rest of their lives.