Border Health Security Act of 2020 This bill modifies the duties and activities of the United States-Mexico Border Health Commission and the Canada-United States Pan-Border Public Health Preparedness Council, and makes other changes to address health issues along the U.S. border. Specifically, the commission must cooperate with the council where appropriate, and both the commission and the council must develop strategic plans, work plans, evaluations, and reports concerning their activities. The Department of Health and Human Services must award grants to (1) eligible entities in border areas to address recommendations of the commission and the council to improve the health of border area residents; and (2) trauma centers, public health entities, and state, local, or tribal entities for infectious disease surveillance activities in border areas. In addition, the Office of the Assistant Secretary for Preparedness and Response may coordinate with the Department of Homeland Security in establishing a system that alerts clinicians and public health officials to emerging health threats in border areas. The Government Accountability Office must evaluate commission and council activities every two years.