Union members picket outside the Trump Taj Mahal casino in Atlantic City, N.J., on Friday. Photo: Associated Press

Workers at the Trump Taj Mahal casino and hotel in Atlantic City walked off the job early Friday morning and took to picket lines after contract negotiations broke down this week.

Unite Here Local 54, the union representing nearly 1,000 Taj workers, organized the strike after determining that the casino, owned by activist investor Carl Icahn, wasn’t offering a serious proposal to cover their health insurance, union spokesman Ben Begleiter said.

The company said in the statement that it was bargaining “in good faith” with the union and threatened that the casino might close as a result of the strike.

The Taj wasn’t accepting hotel reservations for the next 10 days on its website as of Friday afternoon. A company official said in the statement that the casino was fully open and welcoming guests through the July 4 holiday weekend and beyond.

Four Atlantic City casinos, including another one owned by Mr. Icahn, avoided a threatened strike after coming to tentative terms with the union earlier this week. The union had earlier determined that it wouldn’t strike the city’s three other casinos. There are eight casinos currently operating in the gambling enclave.

Full health coverage has long been a mainstay for casino workers, who have traditionally been able to achieve stable, middle-class service-industry jobs in gambling hot spots such as Atlantic City and Las Vegas.

But that has broken down somewhat, especially in Atlantic City, whose casino industry has been hit hard by bankruptcies, private-equity ownership and the expansion of gambling elsewhere in the mid-Atlantic region. Four casinos in Atlantic City have closed in recent years.

The Taj, opened by Donald Trump in 1990, emerged from bankruptcy, filed in 2014, earlier this year with Mr. Icahn taking over ownership of the company, called Trump Entertainment Resorts Inc. Mr. Trump lost control of the casino more than a decade ago. The latest bankruptcy was the casino’s fourth.

Mr. Icahn won in bankruptcy court approval to eliminate union pensions and health-insurance benefits, arguing that restoring them would force him to close the casino. The decision was upheld earlier this year by a federal appeals court.

In a recent survey, the union found that one-third of Taj workers had no health insurance, while 50% were on taxpayer-subsidized insurance, Mr. Begleiter of the union said.

“Historically there has been free family health care and workers have given up wages over the years to maintain that,” Mr. Begleiter said. That is why, he said, “when Taj took [the health insurance] away it was like a double slap in the face. Health insurance has always been the No. 1 issue among our workers.”

Taj Chief Executive Tony Rodio said in a statement that the company had offered health benefits that included employer contributions.

“The employees of the Taj Bargaining Committee seem hellbent on trying to close this property and killing the jobs and livelihood of the other Taj employees including their own union members and members of other unions notwithstanding the fact that Taj ownership has presented good faith concrete progressive proposals to restore certain employee benefits including contributions toward employee health care,” Mr. Rodio said.

“They are hurting their own and everybody else during the busiest time of the year.”

The health-care situation at the Taj had complicated negotiations with other casinos because the union contracts allow other casinos to adopt the contract that is most advantageous to the employer, several people involved in negotiations for casinos and the union said.

Despite that, the union earlier in the week came to tentative contract agreements with Caesars Entertainment Corp., which owns three casinos in Atlantic City, and the Tropicana casino, also owned by Mr. Icahn. The union decided not to strike those properties as a result. Those tentative deals still need to be presented to the union’s workers, union officials said. They include maintaining full free health coverage, Mr. Begleiter said.

The previous strike in Atlantic City occurred in 2004, when 10,000 workers went on strike against seven Atlantic City casinos, and lasted 34 days.

Write to Alexandra Berzon at alexandra.berzon@wsj.com