Sonic booms aren't created by the aircraft's engine noise. They're created by the aircraft moving through the air. The aircraft has a non-zero volume (an aircraft with zero volume is like the emperor's new clothes; you can argue all you want how much design and engineering work it took but all anyone will see is an empty spot on the flight line), meaning it must displace air as the plane passes through air. This displaced air equalizes with the surrounding air in an expanding pressure wave, which moves at the speed of sound in all directions.
As your hypothetical silent aircraft approaches the speed of sound, the air being pushed aside can no longer equalize by moving out directly in front of the aircraft. This causes a single strong compression wave to build in front of the aircraft. Similar waves have already started to build over the top of the wing surfaces, as the airflow over the leading edge of the wing (by design intended to make the air move faster thus lowering its pressure and producing lift) is temporarily supersonic before friction causes it to settle back below sonic speed; at this transition point between super- and sub-sonic, air still moving supersonically "backs up" against the subsonic airflow, creating a wall of relatively high-density air.
At this point, an observer would already hear a small sonic boom from some combination of these compression waves. However, the boom becomes much more pronounced as the entire aircraft transitions beyond Mach 1. At this point the "wall" in front of the aircraft becomes more of a "cone" as the air can no longer build up directly in front of the jet and is instead violently pushed aside at any local "disconnect" in the angle presented to the oncoming airflow. This sudden compression produces a strong shockwave "wake" which is what is heard as a sonic boom.
As a perfect example of this, there actually is (was) a vehicle that spends a very critical portion of its flight under zero engine power, effectively silent in that regard, at not only supersonic but hypersonic speeds. This vehicle, as shown in the linked YouTube, produces a characteristic "double sonic boom", proof that even a "silent" aircraft isn't silent when it's supersonic. Can you guess what it is?
The noise of an aircraft's engines could, conceivably, add a little extra to the sonic boom; the biggest difference is that immediately following the sonic boom, if you were close enough, you'd also get the engine noise, which can be deafening all by itself at full throttle up to 100 feet away.