Such a statistic is not only incorrect out the gate but also demonstrates a problem that has literally nothing to do with taxes at all.
First, the idea that the poor pay zero taxes is in general a falsehood. That 'fact' looks at federal income taxes ignoring the massive taxes that those poor do pay - social security, Medicare/Medicaid, the taxes that your employer pays to employ you such as the payroll tax and a ton of other hidden taxes that exist within the code to hoodwink the poor into thinking that the system is giving them something that it is not. That does not even get into other tax items like the gas tax or the sales tax that proportionately hit the lower incomes far more than the upper incomes. That is why a flat tax is, IMHO, a NECESSITY - a tax that is even across the board where everyone knows EXACTLY what the government is costing them.
Further, the problem that such a 'fact' points out is NOT that the wealthy are taxed to much, the poor are not taxed enough or that there is a disparity in the tax system in general. The problem it highlights is that in a tax system that is based on earnings it shows that the top 20% are essentially taking almost ALL the earnings in the nation. That is a clear problem. There is no reason that the top 20% are taking such massive percentages of the overall wealth generated in the nation each year. Clearly we have serious systemic problems that encourages that kind of pooling of the wealth.