Is it possible to enroll my children in the government health insurance / social security I pay for myself every month as a employee with a work permit? Or is there another solution for my kids except private health insurance? 

No, you cannot enroll your children. Dependents are not covered, only the employee.

 

If your children are Thai citizens then they are automatically covered under the universal health scheme (AKA 30 baht scheme) though only at government hospital.

 

If your children are not Thai, or if you want the option of private care then indeed private insurance is your only option.