Being fired will not show up on a background check. But if contacted by a potential employer, Starbucks can confirm that you worked for them and that you are not elligible for rehire.

i think most jobs you would apply to will only see the jobs you put on your resume. if you get fired i would recommend not to list any job you were fired from as they typically ask for contact information for that job and if they call to ask about your employment there they will most likely be told you were fired.