X-ray Fourier-transform ghost imaging has the potential to achieve tabletop nanoscale microscopy. However, due to the limited luminous flux in practical applications, the imaging signal-to-noise ratio is low, which leads to poor image quality. In light of the binary characteristics of the X-ray modulating screen, this paper studies the X-ray Fourier-transform ghost imaging using a super-Rayleigh speckle field to solve the above problem. The theoretical derivation of the X-ray speckle field generated by the binary modulation screen is first carried out. Then, with the speckle contrast and the difference of local contrast as objective functions, the non-dominated sorting genetic algorithm with elite strategy is adopted to optimize the design of the binary modulation screen. Numerical simulation results show that the proposed method can obtain high-contrast X-ray super-Rayleigh speckle fields, with which the Fourier-transform ghost imaging can be realized. As a result, the image visibility can be enhanced, and the image quality can be improved especially at a low signal-to-noise ratio.