The Game of the Century occurred, of course, during the era when freshmen were ineligible for varsity play. A separate freshmen team would be fielded and would compete either against other four-year colleges' freshmen teams or against junior colleges.
As a preliminary to the UH-UCLA varsity game, the Cougar freshmen (coached by assistant Harvey Pate) faced off against Tyler Junior College.
Looking at the historical list of all-time starting line-ups in the 1989-90 Cougar media guide (which I picked up during one of my two years at UH), I can see that among the 1967-68 UH freshmen, only Bob Hall later became a starter for the varsity, in 1970-71. One other member of the 1967-68 Cougar frosh later earned a varsity letter, Randy Kight.
Given that a school can only award a few scholarships per year, I would hypothesize that only a small number of freshman players would have been eyed from the outset as potential varsity conributors down the line. The rest presumably would have been walk-ons.
One would think that, had Tyler JC possessed any potential NCAA players, the Astrodome the night of the Game of the Century would have been a superb recruiting ground. Indeed, Tyler's Poo Welch transferred to UH and became a two-year starter for the Cougars (1969-70 and 1970-71). A biographical sketch of Welch can be accessed by clicking here and then, when the page comes up, scrolling down to the bottom.