The evolutionary scenario concerning the production and the cooling of He white dwarfs is revisited and discussed. We show that, for a given star metallicity, the minimum mass of H-rich envelopes able to support a red giant structure linearly decreases when the logarithm of the red giant luminosity increases. Increasing the metallicity the mass of this critical envelope decreases. We found that below the upper mass limit for the production of He white dwarfs there exists a rather large interval of masses which, if diffusion is neglected, are expected to cool down quietly, substantially supported by residual pp burning in the stellar envelope and with evolutionary times larger than similar times for CO dwarfs. Under the above quoted assumption, only less massive stars will begin to suffer CNO burning instabilities during the cooling. An increase in the star metallicity acts in the sense of increasing the stability of the structure against the occurrence of CNO flashes. Theoretical constraint concerning the possible occurrence of He dwarfs in galactic globular clusters are finally discussed, showing that even a 10% of quietly cooling dwarf could give a not negligible contribution to white dwarf (WD) cooling sequence to be observed in the clusters.