Asked by CNBC for a comment, Outerwall said it reiterated the statement made on Oct. 4 regarding Jana.
"Outerwall welcomes the opinions of its shareholders and is always open to constructive input toward the goal of enhancing shareholder value. Our Board of Directors and management team regularly review the company's strategic priorities and opportunities, including capital allocation, and assess a variety of strategic options. We are committed to driving value for all Outerwall shareholders and will continue to take actions to achieve this important objective," the statement said.
The stake makes Jana Outerwall's largest shareholder. Rosenstein told "Closing Bell" the company is "gushing with cash," but bearish investors worry management will squander that. He plans to turn things around by convincing management to return cash to shareholders.
Rosenstein said bears also are concerned that Redbox is a dying business, though he thinks it has a "long life."
Jana, which often pressures the companies it invests in to change strategy or sell themselves, last month disclosed a 6.2 percent stake in Safeway. As soon as Jana arrived, management at the grocery chain developed a so-called poison pill, a strategy used by businesses to discourage hostile takeovers by activist investors, typically by attempting to make the stock less attractive.
"That was a little disappointing," Rosenstein said. "It exacerbates the problem. It's a negative signal to their shareholders. I think it's a bad idea for a CEO to do it. I think it actually works cross purposes to what they want to achieve."
—By CNBC's Drew Sandholm. Follow him on Twitter @DrewSandholm