"It is now up to Hostess' BCTGM-represented employees and Frank Hurt, their international president, to decide if they want to call off the strike and save this company, or cause massive financial harm to thousands of employees and their families," Rayburn said.
"Hostess Brands is making a mockery of the labor relations system that has been in place for nearly 100 years," union president Hurt said in a statement earlier this week. "Our members are not just striking for themselves, but for all unionized workers across North America who are covered by collective bargaining agreements."
The union, which represents more than 80,000 industry workers, maintains that the company's policies will bring its members back to workplace standards of the 1950s.
The wind down means the closure of 33 bakeries, 565 distribution centers, approximately 5,500 delivery routes and 570 bakery outlet stores throughout the United States, the company said.
Last Thursday, the Teamsters Union recommended that BCTGM vote by secret ballot over whether workers wanted to continue their strike.
The Teamsters said that the BCTGM members had previously voted "without complete information," with over 90 percent rejecting Hostess' contract offer.
The Teamsters warned that Hostess' Wednesday announcement that it would not be able to operate without the striking workers was "not an empty threat or a negotiating tactic, but the certain outcome if members of the BCTGM continue to strike."
"The BCTGM leaders are putting Teamster members in a horrible position – asking them to support a strike that will put them out of a job when they haven't even asked all their members to go on strike," the Teamsters said in a statement.
ABC News' David Wright and Alex Stone contributed to this report.