  We present forecasts for constraints on deviations from Gaussian distribution
of primordial density perturbations from future X-ray surveys of galaxy
clusters. Our analysis is based on computing the Fisher-Matrix for number
counts and large-scale power spectrum of clusters. We consider a survey with
high-sensitivity and wide-area to detect about 2.5 x 10^5 extended sources.
Based on the self-calibration approach, and including Planck priors in our
analysis, we constrain 9 cosmological parameters and 4 nuisance parameters,
which define the relation between cluster mass and X-ray flux. Because of the
scale dependence of large-scale bias induced by local-shape non-Gaussianity, we
find that the power spectrum provides strong constraints on the non-Gaussianity
f_nl parameter, which complement the stringent constraints on the power
spectrum normalization, \sigma_8, from the number counts. To quantify the joint
constraints on theese two parameters, that specify the timing of structure
formation for a fixed background expansion, we define the figure-of-merit
FoM_SFT = (det[Cov(sigma_8,f_nl)])^{-1/2}. We find that our surveys constrain
deviations from Gaussianity with a precision of \Delta f_nl~10 at 1 \sigma
confidence level, with FoM_SFT~39. We point out that constraints on f_nl are
weakly sensitive to the uncertainties in the knowledge of nuisance parameters.
As an application of non-Gaussian constraints from available data, we analyse
the impact of positive skewness on the occurrence of XMMU-J2235, a massive
distant cluster discovered at z~1.4. We confirm that in a WMAP-7 Gaussian
Lambda CDM cosmology, within the survey volume, ~5x10^{-3} objects like this
are expected to be found: to increase this probability by a factor of at least
10, one needs to evade either the available constraints on f_nl or sigma_8.
