Short answer: NO
Except: Certain Netflix Original Documentaries bear a "Grant of Permission for Educational Screenings". See more below.
Long answer: The Netflix Terms of Use, which you agreed to when you created an account, specifies that the account is "only for your personal, non-commercial use." Netflix is not alone; Amazon Prime, Hulu, and other personal streaming vendors do not grant rights for institutional or educational use.
This is important because licenses overrule copyright exemptions. Showing your personal DVD during class is covered by a specific copyright exemption (Section 110), and showing clips can be covered by fair use (Section 107). However, streaming videos from personal subscription vendors in your classroom when the license prohibits such viewings? There is no copyright exemption for that and the situation is problematic.
This is why the library offers to obtain streaming videos through institutional subscription vendors like Alexander Street Press, Films on Demand, Kanopy, Swank or Docuseek2.