I've never had the opportunity to talk with anyone who considers themselves a Buddhist Christian
A Trappist monk, named Thomas Merton had extensive contact with Buddhist leaders including the Dalai Lama, the Japanese writer D.T. Suzuki, and the Vietnamese monk Thich Nhat Hanh. So much so that when the Dalai Lama comes to the United States he often stays at Merton's old monastery, Abbey of Gethsemani in Kentucky. He wrote extensively (over 70 books) but the two that deal most directly with Zen and Christianity are The Wisdom of the Desert and Zen and the Birds of Appetite.
See also:
Zen Catholicism, Dom Graham (a Benedictine monk), 1964.
Benedict's Dharma, Patrick Henry, Riverhead Books, 2001 - is a reflection of the Rule of St. Benedict by Zen priest Norman Fisher, meditation teacher Joseph Goldstein, professor Judith Simmer-Brown, and Yifa, a nun of the Chinese Buddhist tradition.
As to the Golden Rule, referred to in theological terms as Ethic of Reciprocity, there is nothing inherently Christian about it. Rabbi Hillel, one of the foremost scholars of the Torah once said: "What is hateful to yourself, do not do to your fellow man. That is the whole Torah; the rest is just commentary. Go and study it."
Bahá'í Faith: "Ascribe not to any soul that which thou wouldst not have ascribed to thee, and say not that which thou doest not." "Blessed is he who preferreth his brother before himself." Baha'u'llah
Brahmanism: "This is the sum of Dharma [duty]: Do naught unto others which would cause you pain if done to you". Mahabharata, 5:1517 "
Buddhism: "Hurt not others in ways that you yourself would find hurtful." Udana-Varga 5:18
Confucianism: "Do not do to others what you do not want them to do to you" Analects 15:23
"Tse-kung asked, 'Is there one word that can serve as a principle of conduct for life?' Confucius replied, 'It is the word 'shu' -- reciprocity. Do not impose on others what you yourself do not desire.'" Doctrine of the Mean 13.3
"Try your best to treat others as you would wish to be treated yourself, and you will find that this is the shortest way to benevolence." Mencius VII.A.4
Ancient Egyptian: (the original dates to circa 1800 BCE, and is the oldest version of this rule: "Do for one who may do for you, that you may cause him thus to do."
Hinduism: "This is the sum of duty: do not do to others what would cause pain if done to you. Mahabharata 5:1517
Islam: "None of you [truly] believes until he wishes for his brother what he wishes for himself." Number 13 of Imam "Al-Nawawi's Forty Hadiths." 3
Jainism: "Therefore, neither does he [a sage] cause violence to others nor does he make others do so." Acarangasutra 5.101-2.
"A man should wander about treating all creatures as he himself would be treated. "Sutrakritanga 1.11.33
Judaism: "...thou shalt love thy neighbor as thyself.", Leviticus 19:18
And it goes on...