The liner notes of the deluxe edition of Headquarters list simply "electric piano" in the credits (played by Peter Tork), but I think the liner notes do reveal the specific electric piano, if only inadvertently. The first two pages of the booklet show a picture of Peter Tork and Michael Nesmith in RCA Studios in March 1967:
(Because I couldn't find the same picture on the internet, I scanned my copy of the liner notes, so the quality may not be the best.)
To the left of Nesmith, behind his pedal steel guitar, there's an electric piano. The date of this picture (March 1967) matches the recording date for "Early Morning Blues and Greens" given in the liner notes (22 March 1967 "& other dates"), so it seems likely that this is the very electric piano used for the track. I compared this instrument to various pictures on the internet, and I think it's a Wurlitzer 140B.
Here are some pictures of a Wurlitzer 140B for comparison: