I don't see the reason why you might want to convert curves to outlines but I'd recommend you export your document to PDF first and do this in Acrobat using the Print production > Preflight tool. Underlines don't disappear there.
Underlines do not convert to outlines, sorry to say. If this doesn't occur frequently, you could manually put a line, I suppose. But why are you outlining the text in the first place? If going to a printing company, they should not be requiring this, unless they have very antique equipment.