Economic Effects of Presidential Candidate Proposal: Pay the same % in taxes no matter what you gain
Looking at the matter from a strictly positive and not normative or political view, then 

1) Ignoring "Laffer-curve" effects, bleeding the Government treasury may not happen, if the flat tax rate is set to a suitable level (the specific politician has exactly in mind to bleed the federal government treasury, but that's discussing politics) 

2) As regards income inequality, if indeed the current taxation system is effectively progressive and not just nominally, then it will indeed increase inequality, since this flat tax rate should be expected to lie above the current minimum tax rate implemented, and below the current maximum tax rate implemented.

A plausible anticipated effect is that lower incomes will reduce their saving (i.e. we should expect that the loss in disposable income will hurt savings more rather than consumption), while higher incomes, which will now have more disposable income, will increase savings. While the overall effect on the "consumption/savings" mix may be ambiguous, this will affect also the distribution of wealth, increasing also wealth inequality.

Is inequality bad for economic performance? Check some recent work by S. Turnovsky, that models links between inequality and growth.