What would happen if the entire blade on a turbojet engine went supersonic?
Not much if designed for it, but efficiency will suffer. Also, if the blade is not designed for the loads, the all-supersonic flow could overload it.

How would you achieve the desired all-supersonic flow? By cooling the intake temperature or increasing the rotation speed, of course. Cooling is less practical, so lets assume you run the compressor faster. This would mean higher centrifugal loads, and the increased dynamic pressure would drive up aerodynamic blade loads with the square of the speed increase.

Also note that compressor blades for subsonic flow are highly cambered. Supersonic blades would better be uncambered. Thus, the blade would only function well if both its aerodynamics and its strength are designed for the full supersonic speed. A regular blade would perform poorly and probably fail. However, even an optimized blade would show lower efficiency due to the inherently higher aerodynamic losses in supersonic flow.