WASHINGTON — When two corporations owned by Christian families, one of them Mennonite, filed suit over the Affordable Care Act, they described their complaint in simple terms: The government is forcing them to either break the law or betray their faith.

But at the Supreme Court on March 25, nothing was so clear as the justices explored the murky territory where an employer’s religious rights collide with the interests of its employees or the government.

On the one side is the Hobby Lobby arts-and-crafts chain and Conestoga Wood Specialties cabinetry company. On the other is the federal government, which argues that the landmark 2010 health care law gives women a statutory right to choose among 20 methods of birth control.

The court, judging from the justices’ questions, is clearly divided on this potential earthquake of a religious rights case. It could be yet another instance where Justice Anthony Kennedy provides the swing vote — in this case whether a corporation has religious rights, and whether those rights have been trampled.

Hobby Lobby and Conestoga object to paying for the full range of birth control drugs and devices as required by the Affordable Care Act. To them, a handful of the methods they must cover could cause abortion.

Justice Elena Kagan took up the government’s case from the bench, avowing that the families’ religious convictions were beyond doubt. But she suggested that exempting them from the law would open the door to exemptions for a slew of employers who didn’t want to cover a host of medical services — from vaccinations to blood transfusions.

Kagan suggested the corporate owners had another choice: “Hobby Lobby could choose not to provide insurance.” Under the health care law, the companies would have to pay a tax instead, but it would be comparable to the costs of insurance, she said.

Paul D. Clement, the lawyer for the companies, countered that such a tax would better be described as a penalty and that the corporations would suffer for it, having to raise wages to compensate for their lack of a health care plan.

Clement leaned heavily on the language of the other law that’s central to this case: the 20-year-old Religious Freedom Restoration Act, which requires the government to meet the most stringent legal test before impinging on religious rights.

Solicitor General Donald B. Verrilli, who argued the government’s case, tried to poke holes in Clement’s position by questioning whether the companies have a right to sue under RFRA in the first place. After all, he argued, RFRA refers to a “person,” and a company is not a person and does not have religious rights. If the court decided that companies do have religious rights, it would be “a vast expansion of what Congress thought it was doing” in passing RFRA in 1993.

Looming in the background of the case was the court’s 2010 Citizens United ruling, in which the court decided that corporations have free speech rights. Supporters of the Green family behind Hobby Lobby and the Hahn family behind Conestoga have argued that surely other First Amendment rights extend to corporations as well.