This likely has to do with folding of the results due to the Nyquist behavior (http://en.wikipedia.org/wiki/Nyquist_frequency)

You can see it occur in a Manipulate like this

Manipulate[F = Table[Piecewise[{{1, Abs[i] < x}}], {i, -10, 10, 1}];
 A = Fourier[F]; GraphicsRow[{ListPlot[Abs[F]], ListPlot[Abs[A]]}],
 {x, 1, 10, 1}]

or this

Manipulate[
 F = Table[Piecewise[{{1, i < x}}], {i, -10, 10, 1}];
 A = Fourier[F]; GraphicsRow[{ListPlot[Abs[F]], ListPlot[Abs[A]]}],
 {x, 1, 10, 1}]

Note that since you've chose a very large number of points it appears that function is multivlaued. However this is just an illusion because the point values are so closely spaced and they are alternating between the two "curves". You can see this by choosing a much smaller number of points as in

F = Table[Piecewise[{{1, Abs[i] < 5}}], {i, -10, 10, 1}];
A = Fourier[F]; ListPlot[Abs[A]]