Was the Term 'the Hebrews' a Special Christian Terminology?

The reason I ask this is that I was reading about the sect the Philosophumena calls 'the Peratae' and realize at once how closely related this is to the traditional Jewish etymology of the term 'Hebrew.'  In the Philosophumena, Hippolytus explains the term Peratae to derive from the Greek peran - the word used in the Septuagint to translate the Hebrew word eber or 'beyond,' 'to cross.'  Hippolytus gives a mystical explanation of why the sect calls themselves 'Peratae' but it ultimately comes down to the experience of the ancient Israelites crossing the sea.  The very same explanation is traditionally given in the rabbinic literature for the term 'Hebrew' - "why does he call them Ibriim? (Ex 3:18)  Because they passed (shibri) the sea." (Pesikta Rabbati s. 24)

The point is that the 'Gospel According to the Hebrews' and 'the Epistle to the Hebrews' might not simply be a statement of traditional ethnic identity.  There might have been some significance within the Christian community to being identified as a 'Hebrew' cf. the beginning of the Gospel of Philip - "A Hebrew makes another Hebrew, and such a person is called "proselyte" ... When we were Hebrews, we were orphans and had only our mother, but when we became Christians, we had both father and mother."  I wonder if those who were in the beginning stages of becoming Christians were called 'Hebrews' or Peratae.