You don't need to take any high school classes. If you are under 18 you might need to take the California proficiency test (I'm don't know if that is the same as a GED or not) for the CA school. You should probably check with the schools
You do need to take the list of prereqs but these are college level classes so high school level classes would not fill the requirement - unless you took them as AP classes and passed the AP test or took them as a dual-enrollment student (actually enrolled in a college and in the college class while you were high school student). Once in a while, but rarely, a prereq will specifiy that a high school class will fill it. More often, the prereq listed will have a prereq that could be either high school or college.
Some schools require you finish all the prereqs before you can apply to the nursing program, most recommend that but some expect students to take some of the nursing classes while they take some of the prereqs.
There may (and probably are) other classes that are needed to graduate but aren't listed by the school. These are general education (gen ed) classes and can be taken anytime. Most people take them before entering the nursing program because the nursing classes don't leave much extra time. Gen Ed usually include things like a physical education class, history or social studies, a fine art, sometimes a foreign language. They will be listed under graduation requirements with a link from either the main page or the academics page (or put it into the search bar of the school website). Your CA link doesn't show them (or the college admission requirements) because it is all about the nursing program, not the college in general.