Jet engine powered backpacks were developed in the 1960s or 1970s. I remember reading articles on them at the time. There was a practical limitation in how much weight the pilot could support with his legs for takeoff and landing. Jet fuel weighs about 7 pounds per gallon and jet engines typically have their highest fuel consumption at low altitudes. The units tested back then typically only carried enough fuel for about 5-10 miles of range. A better solution was the Williams WASP, which used the same jet engine but didn't require the pilot's legs for landing gear. It could carry a lot more fuel but was noisy and considered somewhat hard to control.

Cruise missile engines probably wouldn't be the best alternative for this application. They're only designed for an operational life of a 2-3 hours. A better engine might be the Williams FJ-22.