The origin of acupuncture is Chinese Taoism. Taoism is the philosophical system evolved by Lao-tzu and Chuang-tzu that advocates a life of complete simplicity, naturalness, and non-interference with the course of natural events in order to attain an existence in harmony with the Tao, or life-force. It is closely related to Hsuan Chaio, which is a popular Chinese religion that purports to be based on the doctrines of Lao-tzu, but which is actually highly eclectic in nature and characterized by a pantheon of many gods, superstitions, and the practice of alchemy, divination, and magic.
In this Chinese philosophy/religion there are two principles. The first is the "yin," which is negative, dark, and feminine, and the second is "yang," which is positive, bright, and masculine. The interaction of these two forces is thought to be the guiding influence for the destinies of all creatures and things. One’s fate is under the power of the balance or imbalance of these two forces. Acupuncture is a mechanism practiced by adherents of Taoism that is used to bring the "yin and yang" of the body into harmony with Tao.
While the underlying philosophy and worldview behind acupuncture is decidedly unbiblical, that does not necessarily mean the practice of acupuncture itself is against the teachings of the Bible. Many people have found acupuncture to provide relief from pain and other ailments when all other treatments have failed. The medical community is increasingly recognizing that, in some instances, there are verifiable medical benefits from acupuncture. So, if the practice of acupuncture can be separated from the philosophy/worldview behind acupuncture, perhaps acupuncture is something a Christian can consider. Again, though, extreme caution must be taken to avoid the spiritual aspects behind acupuncture. Most acupuncture practitioners genuinely believe in the Tao/yin-yang philosophy that is at the origin of acupuncture. A Christian should have nothing whatsoever to do with Taoism.
As to the difference between acupuncture and acupressure, with acupressure, instead of needles, pressure is placed upon nerve centers. For instance, there are pressure points said to be in the sole of the foot and the palm of the hand that correspond to other areas of the body. Acupressure would seem to be very similar to deep-tissue massage therapy, where the muscles of the body are exposed to pressure to increase blood flow. However, if acupressure is practiced to bring the body into the harmony of yin and yang, then the same problem arises as with acupuncture. Can the practice be implemented without the philosophy?
The important issue here is separation for the born-again believer from any and all practices that would bring him or her the danger of bondage to counterfeit religions. Ignorance of evil is a danger, and the more we inform ourselves as to the true origin of the Eastern philosophies and practices, the more we see that they are rooted in superstition, occultism, and false religions that are in direct opposition to God’s Word. Can a valuable medical procedure be invented by a non-Christian? Of course! Much of Western medicine has its origin in practices/individuals that were just as unchristian as the developers of acupuncture. Whether or not the origin is explicitly Christian is not the issue. What procedures we subject ourselves to in search of healing/relief from pain is a matter of perspective, discernment, and conviction, not dogmatism.