Actually, that is the Rod of Asclepius. It was the symbol of the ancient cult of this god of healing, which was related to snakes. The Hipoocratic Oath starts: "By Apollo the Healer and Asclepius..."
It may be derived from the treatment of the parasitic worm Dracunculus Medenensis, which entailed dragging it out of the subcutaneous tissue by means of a stick. Alternately, it may have simply been derived from Asclepius' association with snakes, who were present in his sanctuaries and said to aid in healing.
Regardless, this medical symbol is derived from the Greco-Roman medical tradition and is of pagan origin. It predates the Christianisation of the Empire, and likely was strengthened and survived due to the erroneous connection to the Biblical Brazen Serpent Nehusthan.
Incidentally, especcially in the US, some medical organisations use a double snake over a staff with wings. This is the Caduceus, and is a symbol of Hermes. This is actually a symbol of Merchants and was mistakenly applied to the US army medical corps in 1902 by an uneducated top brass officer, to the chagrin and embarrassment of many, subsequently spreading to civilian usage. It is profoundly inappropriate, as Hermes was related to making money and was a psychopomp, taking the dead to Hades.