Veteran STEM Scholarship Enhancement Act This bill revises the Edith Nourse Rogers STEM Scholarship program under the Post-9/11 GI Bill. Specifically, the bill removes a requirement that priority be given to certain eligible students and instead authorizes priority to be given to such students in fiscal years when funds are insufficient to provide scholarships to all eligible individuals. The bill specifies that receipt of such scholarship benefits does not count toward certain limitations that are placed on the period of assistance that is allowed under two or more programs.