A former Commissioner of the New York City Department of Records has contacted me to point out that in fact Greeley "sported one of those whacky neck beards". On examining the portraits of him available on the Internet today (which are probably several orders of magnitude more numerous than the ones I could find in 2002), I see that my correspondent is entirely correct. What I had thought was a ruff or scarf is in fact the man's facial hair, his cheeks and chin shaved but his sideburns extending to meet in front of his neck.
Having realised that the former Commissioner was right and I was wrong, I looked again at the other Democratic candidate beaten by Grant, Horatio Seymour, who lost the 1868 election, and realised that he too had sported a similar arrangement"
I then went back another election and realised that once again the Democratic candidate was more bearded than I had thought. I had been blinded by George McClellen's luxuriant moustache; in fact he did also sport a wee tuft of hair on the front of his chin. It is really clear from this 1862 photograph of him with Lincoln, two years before they stood against each other:
I feel that my statement that there has never yet been an election where more than one candidate had a proper beard remains valid - but only because McClellan's, Seymour's and Greeley's beards are not really proper beards. So I shall have to rephrase that section as well. (And the page could do with a more comprehensive update some time.)
For info - candidates with proper beards won elections in 1864, 1868, 1872, 1876, 1880, and 1888 but lost in 1856, 1884, 1892 and 1916. (Abraham Lincoln was clean-shaven at the time of the 1860 election and grew his beard only later.) Both major candidates in 1904 had moustaches; otherwise the moustached won in 1884, 1892 and 1908, but lost to the bearded in 1880 and 1888, and to the clean-shaven in 1912 (twice), 1944 and 1948. You really needed to know that, eh?