I don't know where the idea that government employees receive free health comes from but I can tell you it's not true. Those in the military receive health benefits along with their families and military retirees get Tri-Care but regular civil service, postal workers, congressmen, ect. purchase their coverage each month.
Each year they receive a list of carriers such as Blue Cross, Mailhandlers, and what options they offer. The employee may select the one(s) they find to best suit their needs. They may also select their payment option as well, bimonthy or monthly. The cost is lower in some cases because of the sheer numbers of employees enrolled in certain plans, but it is Not Free by any means.
In my opinion I think all American families should have the same option as the government employees. With the "public option" being what is now call Government Employee Health Insurance.