What are the stances, if any, of the Eastern Orthodox and Catholic Church on whether Christ was androgynous?
Androgyny: the combination of masculine and feminine characteristics into an ambiguous form.
AFAIK there are no biblical textual references that could interpret Jesus as androgynous, but there are several scholarly arguments for God being androgynous.
Since both the Orthodox and Catholic church adhere to the doctrine of the Trinity, a single God who is both three and one (triune); the Father, Son, and Holy Spirit, "one in essence and undivided", e.g. the Trinity shares one divine essence (ousia); uncreated, immaterial and eternal, any biblical reference to the nature of God is relevant as well.
From Intertextuality in Ugarit and Israel by Johannes Cornelis De Moor, Society for Old Testament Study:
From God and Women: Woman in God's Image and Likeness by John D. Garr, founder and president of Hebraic Christian Global Community
Young's literal translation:
I have kept silent from of old, I keep silent, I refrain myself, As a travailing woman I cry out, I desolate and swallow up together.