jlc has a smart data import tool. as bassblaster said, try loading the eagle design file directly, (assuming it is a complete design) might have to zip first. use the gerber viewer to go over it, it is all done in the quote web page.
if you want to load (eagle) and edit (easyEDA) a design that is converted, I have no idea how well that would work. Would have to run a test.
I looked at the quote form, it basically appears to want gerber files. They do have turorials on how to make gerbers from many programs like Eagle. Doing it from Eagle is sort of a pain, as is the board size limit on the free version (for some projects). I'll see how easy it is to do imports and gerbers using EasyEDA