The rumors have been around for years. There's even the rumor (definitely false) that Kissinger helped the creator of Diplomacy (Allan Calhamer) make the game.

I've been able to find the JFK rumor as quoted in the London Evening Standard.

The Kennedys are said to play it at the White House and I understand the Western Alliance is demanding early assurances that Jack sometimes wins.

-- Angus McGill, London Evening Standard, Mar. 20, 1963

and the Kissinger claim in a magazine article.

...Dr. Henry Kissinger (whose favorite game, quite incidentally, turns out to be Diplomacy.)

-- Gyles Brandreth, "Bits and Pices", Games and Puzzles, May, 1973.

The best newspaper source I could find (that doesn't just bop the factoid on with no detail) was a mid-80s article from The Washington Post:

Allan Calhamer, a Chicagoan who holds the patents, flatly denies this. "Kissinger had nothing to do with it," he says in a tone that indicates he has said the same thing many times. Calhamer invented the game while he was an undergraduate at Harvard in 1953. Kissinger was a grad student at the same time.

There are also rumors among players that Kissinger (like many other Nixon appointees) is an avid Diplomacy player, but nobody seems to know for sure. Kissinger has said nothing publicly on the subject. And would any good Diplomacy player believe him if he did speak out?

One well-known Diplomacy player is Walter Cronkite. This is the man Americans said they trusted more than any other?

(I'll get back to the Nixon reference in a moment.)

Certainly, the publishers Avalon Hill themselves were keen on pressing the idea: "Even professional diplomats, including no less a celebrity than Henry Kissinger, have enjoyed its accuracy and intensity." (The publisher was Games Research from 1961-1975, Avalon Hill starting in 1976.)

Kissinger did talk about board games once in his book On China, but not about Diplomacy; about how the Chinese play weiqi (go), the US plays chess. His claim was that weiqi helps with strategic flexibility, and chess helps with single-mindedness. With that kind of analogy it is possible he might appreciate Diplomacy.

If I just go with my historian's nose, though: the London Evening Standard quote was tongue-in-cheek, and the magazine reference was offhand and likely based on word of mouth; neither is reliable.

...

OK, but what about the Nixon reference? Woodward and Bernstein's book The Final Days mentions the board game:

As the second Nixon term progressed, David [Eisenhower] played games more and more. ... Sometimes he could convince Julie [Nixon's daughter] to play. David and Julie also played a lot of bridge. And David loved the board game Diplomacy, which he always won.

This is a substantial enough reference I'm willing to lift up the "Nixon appointee" reference to "plausible" (which of course, includes Kissinger).

...

Buchanan, W. (Sep-Oct 1974). Diplomacy World #5.

McLellan, J. (2 June 1986.) Lying and Cheating by the Rules. The Washington Post.

Woodward, B. & Bernstein, C. (2013.) The Final Days. Simon and Schuster.