Pope John XXIII is quoted as stating that among the rights to belong to every person, Is the "freedom to form a family." Given the Church's stance on same-sex unions, and other non-biblical family structures, isn't his statement too vague?

As with any quote, historical context is important. Pope John XXIII lived in an era when single‐parent families, and cohabiting couples, were rare. And same-sex unions were inconceivable. Back in the late 1950s and early 60s, "family" meant a married father and a mother, and children. There was still a basic moral consensus, which could be presumed in using phrases such as "form a family." 
 
Today, this is gone, and we must be much more specific. Thus, in reading Pope John XXIII, we must adjust to the context in which he spoke, and cannot reasonably demand the precision that is necessary today. Neither would it be reasonable for our opponents on the marriage question to read into these remarks an approval for the current situation.