I'm assuming by "general education", you're referring to a liberal arts education - a course of study that is broad across multiple content areas, rather than deep into one particular content area.

In terms of when and why did all American universities began to require a liberal arts education - the simplest answer is that they didn't and they don't. There are over 5000 institutions of tertiary education in America - some of which do offer a liberal arts program, or general education requirement, but most colleges and universities offer a variety of courses, many of which are very specific (including long distance truck driving, environmental studies, farming/agriculture, medicine, law, etc.)

The colony colleges [Harvard, Collegiate School (Yale), College of New Jersey (Princeton), King's College (Columbia), University of Pennsylvania, College of Rhode Island (Brown), Dartmouth, College of William & Mary, and Queen's College (Rutgers)] offered what was known as a classical curriculum - Greek and/or Latin, some math, and some sciences. This was basically the earliest version of the American liberal arts education. At the time, it was the sentiment that a young man would best be prepared for his chosen career if he was well-versed in the classics. As the country expanded, and with it the nature of professions and careers, colleges and universities began to specialize for a variety of reasons, including attracting students or establishing a particular reputation.