No, they were not all Eagle Scouts. While at least forty astronauts earned the rank as a youth Eagle scouts, in fact only two Apollo moonwalkers did: Neil Armstrong and Charles Duke.
But most were Boy Scouts, with only James Irwin not being a scout at all. The ranks earned by the 12 moonwalkers (sorted by rank, then by Apollo mission):
- Eagle Scouts: Neil Armstrong (Apollo 11) and Charles Duke (Apollo 16)
- Life Scouts: Edgar Mitchell (Apollo 14) and David Scott (Apollo 15)
- First Class Scouts: Alan Bean (Apollo 12) and Alan Shepard (Apollo 14)
- Second Class Scouts: John Young (Apollo 16) and Eugene Cernan (Apollo 17)
- Tenderfoot Scouts: Edwin "Buzz" Aldrin (Apollo 11) and Harrison Schmitt (Apollo 17)
- Cub Scouts: Charles "Pete" Conrad (Apollo 12)
- Not A Scout: James Irwin (Apollo 15)
Boy Scout rank badges: Scout, Tenderfoot, Second Class, First Class, Star, Life, Eagle (image source: Wikipedia)
The Bald Eagle is however the USA's National Emblem and often depicted on badges, pins, buttons, insignia, and other emblems, so chances are they all wore on their person "The Eagle" in one form or another while walking on the Moon.
And the first Lunar Module (LM) to ever have landed on the lunar surface during the Apollo 11 mission, with Neil Armstrong and Edwin "Buzz" Aldrin onboard being the first and the second moonwalkers, has been named Eagle. Perhaps the most memorable mention of the module's name is described in Wikipedia as:
Armstrong acknowledged Aldrin's completion of the post landing
checklist with "Engine arm is off," before responding to Duke with the
words, "Houston, Tranquility Base here. The Eagle has landed."
Armstrong's unrehearsed change of call sign from "Eagle" to
"Tranquility Base" emphasized to listeners that landing was complete
and successful. Duke mispronounced his reply as he expressed the
relief at Mission Control: "Roger, Twan-- Tranquility, we copy you on
the ground. You got a bunch of guys about to turn blue. We're
breathing again. Thanks a lot.
Additionally, the Apollo 11 insignia, designed by Michael Collins, the Command Module (CM) pilot of that mission, also depicted an eagle with an olive branch in its talons:
So in the strictest sense of the title of your question alone, it might have as well been that so far, only Eagles have walked on the Moon.