Do you have any reference for that, or do you mean Harrington & Richardson? The latter's monogram is H & R with a target in the middle and 1871 underneath.
Also, the order in which the letters on the button read is H F R (according to Waterbury themselves) not H R F, so it wouldn't make sense for your suggestion.
I have put the question about what the letters might stand for to the Waterbury Button Company, but haven't yet received an answer. I'll post it here if/when I get it.