C550 is a JAR 25 aircraft therefore you must have an ATPL to sit left seat.
CJ1,2,3, Premier 1 etc are either JAR23 or 'Commuter Category' therefore you can captain them with just a CPL. Being jets you will have to be two crew for public transport operations or you can fly single pilot for private ops (insurance dependant).
There is alot of info on this in the Bizjets forum. Good luck.