Well, on that the use was certainly more diffuse, and Gygax didn't originate the term. The idea for "dungeons" came specifically from Arneson. It was his idea to have the dungeons underneath the castle as a place for the players (using individual characters) to adventure, and then he showed it off to Gygax. Blackmoor, Greyhawk, et al.
And so I find that double unlikely- first, the idea of "dungeons" beneath a castle had widespread usage prior to Tolkien. Second, Gygax didn't "borrow" it from Tolkien. He used it because of Arneson. It would be an example, unless we had more, of people reading Tolkien back into D&D.