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We present the X-ray analysis of the largest flux-limited complete sample of blazar candidates at z>4 selected from the Cosmic Lens All Sky Survey (CLASS). After obtaining a nearly complete (24/25) X-ray coverage of the sample (from Swift-XRT, XMM-Newton and Chandra), we analysed the spectra in order to identify the bona-fide blazars. We classified the sources based on the shape of their Spectral Energy Distributions (SEDs) and, in particular, on the flatness of the X-ray emission and its intensity compared to the optical one. We then compared these high-z blazars with a blazar sample selected at lower redshifts (z~1). We found a significant difference in the X-ray-to-radio luminosity ratios, with the CLASS blazars having a mean ratio 2.4+/-0.5 times larger than low-z blazars. We tentatively interpret this evolution as due to the interaction of the electrons of the jet with the Cosmic Microwave Background (CMB) photons, which is expected to boost the observed X-ray emission at high redshifts. Such a dependence has been already observed in highly radio-loud AGNs in the recent literature. This is the first time it is observed using a statistically complete radio flux limited sample of blazars. We have then evaluated whether this effect could explain the differences in the cosmological evolution recently found between radio and X-ray selected samples of blazars. We found that the simple version of this model is not able to solve the tension between the two evolutionary results., Comment: 15 pages, 30 figures, 3 tables, accepted for publication in MNRAS