No, human eyes do not shine at night in the same way that some animals' eyes do.
The Absence of Tapetum Lucidum
The reason human eyes don't shine is the lack of a tapetum lucidum. This is a reflective layer behind the retina found in many nocturnal animals, such as cats and dogs. The tapetum lucidum reflects light back through the retina, enhancing night vision. Humans do not possess this layer.
Therefore, shining a flashlight into a person's eyes at night will not produce the characteristic shine observed in animals with a tapetum lucidum. You will not see a reflection. However, a camera flash, being much brighter, can cause a reflection from the retina itself. This is a different phenomenon than the shine caused by the tapetum lucidum.
Camera Flash vs. Flashlight
The difference in the brightness of a flashlight and a camera flash is crucial. A flashlight is insufficient to cause a noticeable reflection from the human retina. A camera flash, however, is powerful enough to produce a reflection visible in photographs. This is not a "shine" in the typical sense, but rather a reflection of the flash.