New York healthcare workers and teachers who don’t want the COVID-19 vaccine will not only lose their jobs but could also be unable to get unemployment benefits. Thousands of nurses and other hospital workers have been put on leave, slated to be fired, as the state’s vaccination mandate deadline for hospital staff passed on Sept. 27. At least some hospital systems have given their staff some time to still get the needle and return to work before facing termination. In addition, New York City teachers have until Oct. 1 to get at least one shot of the vaccine lest they face a year of unpaid leave with health benefits or termination with a severance package. Once fired, the workers could be left without access to unemployment insurance unless they legitimately asked for a health or religious exemption, according to the state’s Department of Labor (DOL). “Workers in a healthcare facility, nursing …