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?

A chemical engineering degree (or any kind of engineering degree) would make you ridiculously overqualified for those jobs. An engineering degree will allow you to work as an engineer, not a technician or a scientist. Engineering jobs are generally high-skill positions with relatively high salaries. Generally speaking, technician jobs tend to be lower-skill positions, especially a biological or chemical technician.

If you want to work as a technician of some kind, major in the appropriate science. Of course, it’s usually ill-advised to major in biology or chemistry since job prospects are extremely poor with just a Bachelors (and not stellar even with a graduate degree), as they will only qualify you for low-level technician positions with little job security and zero potential for advancement.

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/