Live in Sweden for 12 months or more
Employees who stay in Sweden for more than twelve months are registered in the Swedish Population Register, regardless of nationality. A person who is registered in the Swedish Population Register is entitled to healthcare under the same terms as other residents, that is, at the ordinary fee of the public healthcare system.
More information can be found at Vårdguiden's website: Patient Fees
Normally, if you stay more than twelve months, you are also entitled to the social insurance benefits provided by the Swedish Social Insurance Agency (Försäkringskassan). You are, for example, entitled to sickness, parental and temporary parental benefits.
On arrival in Sweden, you must register with the Swedish Social Insurance Agency by filling in form 5456 "Information required to obtain social insurance cover". The Swedish Social Insurance Agency will process your application and decide whether you are entitled to a particular benefit.
More information can be found at the Swedish Social Insurance Agency's website: Insured in Sweden