Cook County’s health system says it will cut about 300 positions in fiscal 2011. But a nurses union is threatening to strike over the plan. The cash-strapped county’s Health and Hospitals System has been shrinking its payroll for years. Now County Board President Toni Preckwinkle is asking for another $83 million in cuts. That’s not going over well with National Nurses United. The union represents 800 registered nurses whose county contract expired more than two years ago. “We are sick and tired of trying to provide high-quality care in an unsafe and understaffed environment,” said union negotiator Dorothy Ahmad, a nurse in the coronary unit of John H. Stroger Jr. Hospital.