Ensuring Network Security Act This bill expands eligibility and establishes distribution priority for funds provided under the Secure and Trusted Communications Networks Reimbursement Program. The program reimburses providers of advanced communications service for costs associated with replacing communications equipment or services that pose a threat to national security. Specifically, the bill increases from 2 million to 10 million the number of customers that an eligible provider may serve. Further, the bill establishes priority for the allocation of reimbursement funds to applicants, providing that such funds shall be distributed first to applicants that have 2 million or fewer customers; subsequently, to applicants that are accredited public or private noncommercial educational institutions providing their own facilities-based education broadband service; and finally, to certain other applicants that are designated as eligible telecommunications carriers.