It is the government who writes your unemployment checks, but

your ex-employer does contribute to it.

The following is quoted from farmerabc's post above:
"The notice advises the employer about whether the claimant is eligible or not, and whether the employer's account will be charged for benefits paid to the former employee."

When economy is good and jobs are plenty, tricks like this work fine since nobody cares to file for unemployment benefits. But in today's economy, I'll be very very alert if an employer asks you to resign without a cause, since most likely the employer just doesn't want to contribute to your unemployment benefits. Otherwise, they will just lay you off. Of course, if you are able to find a new job quickly without filing for unemployment benefits, then it'll be best for both parties.