2. People/Companies refuse contracts based on their principals
3. Other People/Companies take up those contracts
4. Those Companies post job openings to build said houses
5. People take said jobs, they can refuse if they don't want to work for a company working for the Govt.
6. Houses get built
Sounds like regular capitalism to me, no one is being forced to do something they don't want.
If the government are building the houses surely that's the will of the people?