A regressive tax is a tax paid by the poor at higher rates than the more wealthy. It can also be seen as a tax that impacts the poor much more than the rich. The minimum wage tax fits both definitions of regressive.
Increases in the minimum wage receive considerable support from the notion that employers deliberately underpay workers if they can get away with it. The minimum wage would prevent that, or at least that is the assumption.
It is not surprising that workers earning less that the proposed wage would welcome the increase, not suspecting that they might actually lose their job should the increase occur.
To see that the minimum wage is a regressive tax, again assume that government places a tax on all wages below $20 per hour. All wages above $20 per hour are not taxed; all wages below $20 per hour are taxed at a rate equal to $20 less the actual wage rate. The less paid, the higher the tax rate. This situation clearly meets the first test of a regressive tax, the tax increases as the wealth (wage) decreases.
The minimum wage tax also impacts the poor much more than the rich, meeting the second definition of a regressive tax. The less able persons in society are the younger people, the less trained people, the less talented people, the very aged people. Each of these groups is likely to be unable to command a high wage by ability to perform well in the work place. Each of these groups is more likely to be receiving less than the proposed $20 minimum wage, with the least fortunate being most likely to be in this sub-minimum wage group.
Some readers will protest at this point that the minimum wage workers will get the tax, so they will really be winners! Yes, the government can collect the tax from employers and immediately give it to workers. Government can also require that the tax be paid immediately to workers in their pay check. Both methods of returning the tax to employees are identical in the calculation made by the employer deciding to create a job or not.
By placing a tax on all wages below $20 per hour, government has decided that employers should be penalized for assigning such low value tasks to workers. Government has decided that tasks worth less than $20 per hour should not be done, with the exception that if employers DO decide to perform such a low value task, the employer will be penalized by the difference between actual wage and $20.
Clearly, with the minimum wage, government is setting an economic standard. Each change of the minimum wage is a shift between two standards. The economy can be expected to adjust to each standard as time passes. Those within the economy best able to adjust will prosper, those least able to adjust will fall behind.
As the minimum wage increases, the least able will fall increasingly behind.