It's less about the height of the king and more about the width of the pieces' bases. If I'm not mistaken, the USCF's guideline is that four pawns should be able to fit snugly inside the square. With standard Staunton sets, this tends to work out to squares of about 2.25".
2.37 vs 2.5" squares

The "right" size square is a function of piece height+stem thickness+base width, along with personal taste.
Some prefer a crowded board while others like a spacious feel.
I think I am typical when I prefer a 2.5" square with my 4" Collector. I have a thicker, more bulbous-based 4" Collector II and my 2.5" board is not quite an ideal match. I also have a 2-3/8" board and while the 4" Collector is fine on it, 2.5" squares are the better fit.
I got 2.5 on my solid wood board and 2.25 on my veneer board for my Zagreb 59 3.80" pieces... looks good to my eye, but I was just curious what /if any/ the guidelines were. Thanks all for the good info.

...looks good to my eye...
The rule of thumb is that the king's base is roughly 78% the width of a square.
But your rule is more important.

Twice the diameter of the base of the pawns is a good rule, and that used by FIDE.
http://www.fide.com/component/handbook/?view=article&id=16
Too wide, and the pieces look lost. 2.375" is good for 4" pieces, depending upon the bases of the king and queen. If they're narrow, you might want to consider 2.25.
The standard pieces used by FIDE are 3.75" high, and the square size that's become standard is 55mmx55mm. (2.165")
But the standard pieces FIDE recommend now have a narrow base compared to some of the showier sets sold.
I'm sure it's 2.375" in this photo, with the Sultan Queen. Beautiful set - congrats.
Thanks for all the input fellas... I decided to go with the 2.5" for two reasons... build time was a lot shorter, and I plan to get a 4.4" set down the line that I want to use with this board. I think 2.5" is still with in the guidelines as the King is 1.875 wide with this set, bit ideally I'd have the money for two boards.. 2.375 and 2.5.. thanks all for the input.

...looks good to my eye...
The rule of thumb is that the king's base is roughly 78% the width of a square.
But your rule is more important.
roughly 78% ? :)

...looks good to my eye...
The rule of thumb is that the king's base is roughly 78% the width of a square.
But your rule is more important.
roughly 78% ? :)
lol, yeah, 78% is the rule but it never fits exactly so I added "roughly" :)

lol, yeah, 78% is the rule but it never fits exactly so I added "roughly" :)
HOS says add or subtract 1/8" from that 78%, per taste.
Frank wrote that part of the USCF rulebook so it reads the same just in case I wish take my Collector to that rated blitz tournament in the park with the addicts and mental cases.
I am trying to decide between 2.37 and 2.5" squares for a board I'm having made.. the chessmen are HOS Sulton 4" king and so both sizes work I suppose.. but is there anything I need to know about with regards to size to bias me one way or the other? Is there a "Standard" size board?