Can a dental assistant place my permanent crown?

No...never have any work done by an assistant if there is no Dentist in the office....having said that...If trained and certified, a Dental assistant can remove your temporary crown, clean up the prep tooth and try in the new crown. I dont believe in having assistants adjusting the bite on crowns as well, even though I know it is done all the time. 
Only the Dentist can place the final crown with cement and make the final adjustments, 

No, dental assistant cannot place your permanent crown because it is not safe and legal. Assistant will only help the dentist while doing the procedure but he or she doesn't allowed to perform the whole treatment in the absence of dentist.

It is not legal in Texas.  I am not sure in other states as they all have their own dental practice acts.

If trained and certified, a Dental assistant can remove your temporary crown, clean up the prep tooth and try in the new crown. I dont believe in having assistants adjusting the bite on crowns as well, even though I know it is done all the time. 
Only the Dentist can place the final crown with cement and make the final adjustments, 

It is illegal and unethical for an assistant to place a new crown. If an old crown comes off an assistant can place it with temporary cement but not permanent cement.