CSUF Health Science Major transfer to Nursing BSN

I’m an incoming freshman at CSUF, I didn’t get into the nursing program but I got in as undeclared and switched my major to health science. Would I be allowed to transfer/apply to other CSU BSN programs when I try to switch majors to Nursing?And if that doesn’t work out should I finish my BS in Health Science then transfer to an accelerated CSU BSN program or after two years of undergrad just go to private school?