After a fashion they can. Planes, of all types, have two speeds, air speed and ground speed. For airplanes, ones with fixed, horizontal wings and a forward-thrusting propulsion system, air speed is always in the direction the nose is pointing. Ground speed is a different story.
Once a plane leaves the ground, it is subject to the direction of the air mass it is in. So if the headwind is greater than the air speed, the ground speed will be negative and the plane will appear to go backwards. And, yes, planes, like Cessna's 150 can go that slow with full-flaps, etc.
If you talk about helicopters, it is a different story, they can actually fly backwards with a negative airspeed. That is because the propulsion system is Up, not forward. The tilt of the rotorblades determine direction of travel. I used to have a lot of fun dancing around a field while practicing hovering.
In JKenny's answer, the Harrier, and similar aircraft, the wings actually rotate so that the propulsion system actually points upwards. I don't know, but I guess they tilt the wings back and forth to obtain directional movement.