This sounds like the 1991 novel Russian Spring by Norman Spinrad.
Points that match:
I read a book in the late 90's
about an American boy who becomes a rocket engineer for the ESA?
The only real outer-space planning is in Common Europe, so young American "space cadet" Jerry Reed goes to work in Paris.
He has an accident and some shrapnel lands in his brain and paralyses him then he goes for a ride around the moon in a ship he helped design.I particularly remember that the ships pilots were grateful that he included a joystick to control the ship with so that it felt more like flying a aircraft.
Correct. He works for the ESA and designs spaceships and has an accident.
As he is dying he arranges to have his brain plasticized so that future doctors can bring him back and fix him. His doctor is sad that he can't "Blue Max" him, but must wait 6 minutes after death before removing his brain.
They are unable to save his entire body so they apply preservation techniques to just his head.
If this is the book, in a particularly poignant scene you may remember the following:
The last point fulfills a lifelong dream of his to visit the moon which was fostered in him by his father and his uncle during the original moon landings. They allowed him to stay up to watch by allowing him to eat his favorite ice cream so he would remember it. (he was like 3 years old or about that). The ice cream established a sensory memory. When he went around the moon he thought it tasted like a big bowl of ice cream.