From the contract work I've seen, you hire a contractor, who hires employees to build that wall. The mason building the wall can be fired if he does a bad job, but he isn't responsible for guaranteeing the wall's quality. The contractor is responsible. He supervises the job, and checks on the quality. A GOOD software company has the same thing. The company or supervisor is responsible for the quality of the software, and, if they know what they are doing, have QC set in place to bug-check, offering incentives for finding bugs/creating good software.
It sounds like the boss is using a bad analogy to make sure you think the weight of any bugs falls on your shoulders, not his. You are the brick layer. He is the contractor. You don't sign the contract. He does, or the company does.