Would "Biochemical Engineering" qualify me for these jobs?

I want to work in many different fields of science and I was looking at this major from UC Davis: http://admissions.ucdavis.edu/majors/major_view.cfm?major=ebcl

I wanted to ask would it qualify me for the jobs below?

http://www.bls.gov/ooh/life-physical-and-social-science/forensic-science-technicians.htm
http://www.bls.gov/ooh/life-physical-and-social-science/chemical-technicians.htm
http://www.bls.gov/ooh/life-physical-and-social-science/biological-technicians.htm
http://www.bls.gov/ooh/life-physical-and-social-science/environmental-scientists-and-specialists.htm#tab-1

If not, what if I Dual Majored in Chemistry and Biology?

I would say yes to Forensic and Biological techs. Depending on course work Chemical is an option to. I would hire an Enviro.E or or Environmental science major first for environmental tech jobs.

Northern Michigan University has a comprehensive set of science majors that prepare students specifically for clinical science jobs like you described. http://www.nmu.edu/clinicalsciences/