Actually the most generally accepted theory of the flu is that it began in Kansas and was relatively mild, went to Europe where it mutated/got worse and then was re-introduced in to the USA by some returning ships. It was the "Spanish" flu because it was initially widely reported in the Spanish press which unlike the press in the warring countries, was not censored.
Both sides were hit pretty hard by the flu, it is not clear if made much difference unlike a situation where one side or the other was hit much harder by disease.