The best argument that can be made to connect the GI Bill to the counterculture is to say that the GI Bill led to the wealthy and materialistic society of the 1950s and early 1960s. The counterculture arose as a backlash to the way in which society had become so materialistic.
The GI Bill helped America to become much wealthier than it had been before the war. It did this largely by paying for huge numbers of ex-military personnel to go to college. This allowed them to get good jobs in the booming US post-war economy. The GI Bill also helped to bring suburbanization about. It did this because it provided easy loans for ex-military personnel to buy homes in the new suburbs.
With these changes, America became wealthy and materialistic. The people who had been through World War II (and in some cases the Great Depression) wanted to work hard and get ahead. Their children thought that their lives lacked purpose and “soul.” Because of this, they started the counterculture as a backlash against their parents’ way of life.