For UT Austin, general pre-reqs for a master’s degree: http://www.utexas.edu/student/registrar/catalogs/grad05-07/ch3/ch3a.html
“He or she must also have taken at least twelve semester hours of upper-division undergraduate coursework in the area of the proposed graduate major or must have the consent of the graduate dean. Some areas may require more undergraduate preparation. Students who lack adequate preparation may be admitted to a graduate program on the condition that they complete additional preparatory coursework designated by the graduate adviser.”
This is typical. So that’s a minimum of 4 upper-division courses. It looks like Arizona’s engineering minors require ~3 upper-division courses, so you’d want to add a couple.
http://engineering.arizona.edu/academics/declare-minor
Then you’d have to consult each department for additional requirements. For UT Austin Biomed Engineering, for example: http://www.bme.utexas.edu/graduate-program/admissions/admission-criteria
“Recommended Prerequisite College-level Courses:
Calculus-based physics (two semesters), Differential equations, Statistics, Chemistry, Biology (cell or molecular), Physiology.”
You would have fulfilled these courses through your major/minor. Then it says: “Applicants should possess a B.S. or B.A. degree in an area compatible with our program.” If you had a B.S. in something like Molecular & Cellular Bio with a minor in Mechanical Eng, you’d probably meet the pre-reqs for a master’s program such as that.
Point being, an engineering minor would likely allow you to pursue an M.S. Engineering (that corresponds with your overall academic program), if you wanted to go in that direction, but you would have to be careful with the specific requirements of grad departments.