A few dozen Boeing employees recently took to the picket line to express their frustration and their plan not to comply.

More than 7,000 Boeing employees have applied for religious exemptions and about 1,000 are seeking medical exemptions, people familiar with the matter told Reuters, that amounts to some 6 percent of the planemaker's roughly 125,000 U.S. employees.

The Biden administration has ordered any company that wants to keep getting federal contracts to have all contractor employees fully vaccinated against COVID-19 by Dec. 8th.

That's putting some companies already hampered by a labor shortage in a tough spot.

At aircraft companies Spirit AeroSystems and Textron, nearly half of their roughly 10,000 employees remain unvaccinated, according to a union official.

And at defense contractor Raytheon Technologies, CEO Greg Hayes warned last week that he could lose "several thousand" employees because of the mandate.

Many legal experts say vaccine mandates in the interest of public health are legal.

Commerce Secretary Gina Raimando told CBS Face the Nation that the mandate is for the good of all workers.

"People want to work in a workplace where they feel safe. You see, you know, United Airlines, that was among the first to do the mandate. Their- you know, number of folks applying for jobs is through the roof. The best thing we can do to get people back to work is to make sure everybody is vaccinated."

Companies, however, might be able to avoid mass firing by the December 8th deadline. Under government guidance published on Monday, companies will have flexibility on how to implement the mandate.

In the meantime, some union officials are heading to court over the mandate and another brewing battle - this one over how companies decide which exemption requests are rejected.