A 'red petticoat' was C17th slang for a prostitute

The song Siuil a Rún refers to a lover's enlistment in the Irish Brigade who left Ireland after the Williamite War (1691) to serve in the French Army
The Lichtbob's Lassie is about a woman who wishes to give up a life of relative luxury (feather beds are soft) to follow a soldier of the Light Brigade

Such camp followers were often regarded as no better than prostitutes (in many cases they actually were) and I feel the connection with the verse to dye petticoats red is simply a statement that the intention/wish is to follow a lover into a campaign as a prostitute, if necessary.

However the life of their own that some threads take on can be a lot of fun too :)