Can a convergent nozzle be used for a supersonic jet engine?
Short answer: Yes.

A convergent nozzle will not allow supersonic exit speeds of the combustion gasses, but due to their high temperature their speed of sound is considerably higher that that of the surrounding air. For example, at 700°C the speed of sound in air is 625 m/s. Since thrust is mainly determined by the difference in entry and exit speeds of the air flowing through an engine, a higher speed than flight speed is required for positive thrust. Low supersonic flight speeds are entirely possible with a convergent nozzle.

Long Answer: Better not.

Once the aircraft crosses the sound barrier, drag increases only slowly since the drag coefficient actually drops. If the design is meant to fly supersonically, which implicates a lot of design adaptions, it makes sense to go for higher supersonic speed; however, this requires both an adjustable intake and an adjustable convergent-divergent nozzle. Both increase efficiency, dramatically so at higher Mach numbers. The simple pitot intake of the F-16 is just good enough for Mach 1.6 (above that its efficiency becomes outright awful), but still its F-110 engine has a con-di nozzle.

With an adaptable convergent nozzle the exhaust gasses can be accelerated up to their speed of sound, but not more. A narrowing of the flow path will accelerate subsonic flow, but only to the speed of sound. Only a divergent flow path will then accelerate the flow further to supersonic speed. A convergent nozzle will be shorter and lighter, but means throwing away a good deal of useable energy with the hot exhaust gasses. In most cases it would be more efficient to convert the pressure energy of the exhaust gasses into more thrust by accelerating them to supersonic speed.

Convergent nozzles only make sense if the aircraft is designed only for short, limited supersonic dashes, but will spend almost all flying time at subsonic speeds. Examples are the Northrop F-5E or the Panavia Tornado.