The problem is the public jobs part, not Obama’s “private sector” characterization (which seems like ordinary misspeaking). There are at least two ways to read it: 1) As a Keynesian, Obama’s arguing that if state and local governments could stop laying people off for a while it would have a salutary effect on the economy by boosting demand, the same way tax cuts or public works projects or unemployment benefits boost demand; or 2) Public jobs are equivalent to private jobs, including housing jobs. In a healthy economy we have lots of private jobs, lots of housing jobs, and lots of goverment jobs. They’re all good!
Obama clearly means #1–he’s proposed $35 billion in subsidies to help states rehire “teachers and cops and firefighters,” with a fairly explicit Keynesian rationale. But does he also mean #2? There’s nothing in his press conference to suggest he doesn’t, and lots to suggest he does–the parallelism with the construction industry, for example, in which goverment jobs are treated as just another “sector” in which more employment is better.
The problem is that many voters (myself included) don’t think government jobs are just another sector. We want the number of housing and manufacturing jobs to keep growing–the more the merrier, all things being equal. We don’t want the number government jobs to keep growing, in part because we pay for them without the assurances, offered in a competitive private economy, that we’re getting our moneys worth or that the jobs are necessary at all.