  The Hubble diagram constructed using HII galaxies (HIIGx) and Giant
extragalactic HII regions (GEHR) as standard candles already extends beyond the
current reach of Type Ia SNe. A sample of 156 HIIGx and GEHR sources has been
used previously to compare the predictions of LCDM and R_h=ct, the results of
which suggested that the HIIGx and GEHR sources strongly favour the latter over
the former. But this analysis was based on the application of parametric fits
to the data and the use of information criteria, which disfavour the less
parsimonious models. In this paper, we advance the use of HII sources as
standard candles by utilizing Gaussian processes (GP) to reconstruct the
distance modulus representing these data without the need to pre-assume any
particular model, none of which may in the end actually be the correct
cosmology. In addition, this approach tightly constrains the 1 sigma confidence
region of the reconstructed function, thus providing a better tool with which
to differentiate between competing cosmologies. With this approach, we show
that the Planck concordance model is in tension with the HII data at more than
2.5 sigma, while R_h=ct agrees with the GP reconstruction very well,
particularly at redshifts > 10^{-3}.
