I believe you will find that every state in the union, not to mention the fed gov, requires the EMPLOYER to pay all of the unemployment tax. I am a bookkeeper by profession and currently do my brothers payroll taxes...and both the fedgov and stategov brochures indicate it is illegal to charge or withhold unemployment taxes from the employee.
And now that you know, don't feel badly....the companies are forced by government to fund the periods of time when employees are unemployed. If I recall correctly, fedgov just upped the # of weeks you're eligible to collect.