In Switzerland, health insurance is the employee's responsibility. This insurance is mandatory for any person domiciled in the territory of the Confederation. It is up to the individual to choose a provider. There are over 90 providers recognized by the Federal Office of Public Health (FOPH). Providers must accept any individual without any conditions or waiting period, regardless of their age or health status. The same is not true when applying for supplementary insurance, which is subject to certain conditions.
People from across the border who work in Switzerland have the choice to either buy private health insurance in their country of residence or from a provider in Switzerland.
As for accident insurance, from the start of your assignment your employer is responsible for insuring workplace accidents. If the number of work hours exceeds 8 hours a week on average, the employer insures non-work related accidents as well. If you become unemployed, contact your health insurer to add an accident clause to your insurance contract. Similarly, if you change from an unemployed to employed status, contact your insurer to remove the accident clause from your coverage.
More information on the health insurance on the Federal Office of Public Health website
More information on living and working in Switzerland on the Federal Office for Migration website
More specific information can be found in the brochure entitled "Living and working in Switzerland"