http://admission.universityofcalifornia.edu/paying-for-uc/files/fafsa-tips.pdf says:
That is a little vague, since it does not say one must do that, but just says that it is best to do that.
Most UCs do not consider “level of applicant’s interest”, but UCI and UCM do, according to:
http://www.collegedata.com/cs/data/college/college_pg02_tmpl.jhtml?schoolId=730
http://www.collegedata.com/cs/data/college/college_pg02_tmpl.jhtml?schoolId=2046
So, if one of those is among the 6 UCs, perhaps you can put that school first, then the private schools that consider “level of applicant’s interest”, if you are trying to game the FAFSA order to show interest.