I dont think the engine of the 262 would like supersonic flight. 
Subsonic jet engines work by compressing the air, igniting it, and that air if forces through a turbine on a shaft that runs back to the compressor.  At these subsonic speeds, the air is forced faster through small spaces, much like putting your thumb over the end of a hose.
Supersonic jet engines work differently. At supersonic speeds, the air acts completely opposite of subsonic speeds.  The air will actually slow down and pile up through a narrowed region instead of speeding up and forcing its way through.  The air needs to be slowed down before reaching the compressor.  Early jet engines did this by incorporating a point coming from the center of the compressor ahead of the engine inlet.  As the air hit this point, it piled up on itself and slowed to a subsonic speed.  This air was then put through the engine.  New supersonic engines do not include a compressor, they simply narrow the inlet gradually.  This narrowing compresses the air and then it is combusted.  
From a jet engine standpoint the plane could not do it.  Maybe in a dive, but the engines themselves would cause huge amounts of drag (air piling up). and would add no additional thrust.  
I think it could probably hit the transonic range.