Many officers of startups do not make a salary, but they are actively working for their company. While they do have options, these options may not be worth much of anything in the beginning. If the startup is winding down and they are actively looking for employment, would their status for an existing company disqualify them from benefits?
Unemployment benefits are generally a percentage of your wage and salary income for a specified time period, and any percentage of a $0 salary is zero. Also, if you are still a CEO, you are not unemployed. So, no, the CEO would not qualify for unemployment benefits.