Taking a screenshot will not work as far as I know. Unfortunately you need the actual file. Losing everything is a bummer but next time remember to save!
Edit: Although if your friend has the map and you simply want it, he/she can email the files to you so you can get in. If you only sent him/her the files after you exported, you won't be able to edit, but if you sent him/her the actual .png heightmap file, you can get it back that way. Best of luck.