Not sure about the Haxe way, but the easiest solution will probably be to use the Flex SDK to generate a font library SWF.
FlashDevelop includes a "Font Library" AS3 class Template which will be available in AS3 projects. You can create an Empty AS3 project in your Haxe project folder.
This template class offers lots of information on how to embed fonts - make sure you embed each font variant (regular, bold, italic, bold-italic) as required and give them all the same "font family" name that you'll use as the font name in the TextField.
You can copy this AS3 class in your Haxe project (and delete the AS3 project) and build it using the Quick Build (Ctrl+F8) feature.
The resulting SWF can be loaded or embedded - but it must have been executed first for the fonts to be available.