The entitlement mentality starts by seeing one's parents transfer their responsibilities to the government. That’s why we see three and four generations in government housing, on government food stamps and on relief. Kids who grow up in the projects think they are entitled to housing and food. Kids who grow up in government school think they are entitled to an education. From there it is an easy step to thinking they are entitled to healthcare, medication, etc.
Government schools are the first socialist institutions most kids come in contact with. They learn that they can pass off their responsibilities to the government and force their neighbors to pay for it.
No matter what you try to tell your children about responsibility and the free market, your example is showing them that socialism is and acceptable alternative to personal responsibility and capitalism. Aside from all the other problems with government schools this is their worst element in my opinion.