There was some discussion of a Thielert installation in a Cozy some years ago, but I don't think that happened.
There have been a few turbojet Cozy's or Long-EZ's built and flown.
https://www.eaa.org/en/airventure/ea...-at-airventure
The Shockwave installation never took off, but also had a few installations, I believe.
https://eaa1541.org/members/what-our...ccis-cozy-jet/
The Jetcruiser 450, while not a Rutan design, was a turboprop canard single, with an Rolls Royce engine, later re-imagined with a P&W, and in theory, to be produced in kit form in California. It was to be a production aircraft with Part 23 certification. Presently the certificate owner is looking at going electric.
XCOR corporation had a one-off rocket powered Long-EZ (EZ-Rocket) which was actually piloted by Burt Rutan, using two liquid-fueled rocket motors.
https://www.space.com/1834-xcor-rock...cord-book.html
Deltahawk promised a 200 hp diesel engine for a long time, and eventually put some in UAV's, and had an installation in a Cirrus. Their plan from very early-on was to put one in a Velocity, and they did announce an intent to produce an installation for the Cozy MkIV. They were a bit on the pricy side; closer to seventy grand.
The Raptor was a cozy-esque aircraft that did fly using a diesel engine (Audi), and there was some talk of a P2M J3E diesel, but after the prototype crashed in a field during a ferry flight, the project ended. There's a resurrection looking at electric right now, as the Raptor NG.
https://youtu.be/hxFxxFAEv18. The Youtube videos show it flying again, with a diesel engine.
There a ducted fan Long-EZ out there, too. The Duckt used a Mazda rotary. Not diesel, but automotive fuel.
http://mickaircraft.com/duckt.htm