Siddhartha Gautama, the founder of Buddhism, said that what he was teaching had nothing to do with religion and that he would be wrongly deified after his death. He was right. His corpse had barely grown cold before he was canonized and his teachings were alloyed with the existing religions of the time and region. Today those alloyed religions include Hinduism, Shintoism, Animism, Christianity, and even atheism (i.e. no-religion).
Even early on, however, there were followers of Siddhartha who worried about this. Zen arose partly as a form of Buddhist fundamentalism – a return to the “Bodhi Tree roots” (so to speak). Zen stresses direct enlightenment primarily through meditation and secondarily through teaching (the mix depends on the school). Enlightenment is impossible to define in words, and it’s not soteriological like the Christian concept of salvation. Enlightenment in Zen means something more like “direct perception” or “immediate existence.”
So the answer to the question “Can a person be a Christian and a Buddhist” may have different meanings, depending on what one means by Buddhist. If one means Buddhism alloyed with Hinduism, then one can be a Christian and a Buddhist to the same extent that one can be a Christian and a Hindu (etc.).
If by Buddhism one means Zen or other forms of fundamentalist Buddhism, then one can be a Christian and a Buddhist to the same extent one can be a Hindu and a Buddhist, a Shintoist and a Buddhist, an atheist/materialist and a Buddhist, etc. Zen and fundamental Buddhism require no theological input and no theological commitment (in fact, most schools warn against it). Any theological response to enlightenment and practice are considered part of the individual student’s experience and not part of Zen itself.