Gene Synthesis Safety and Security Act This bill requires the Department of Health and Human Services (HHS) to update guidance to address the potential misuse of gene synthesis products (i.e., manufactured genetic products) to obtain biological agents or toxins of concern. The guidance must include recommendations relating to screening protocols, recordkeeping, technological safeguards, and other specified information. HHS must evaluate the current landscape of gene synthesis manufacturers and related technology to inform the guidance. HHS must also solicit public input to develop a list of genetic sequences of concern.