Please help, MacBook Pro eligible for repair program will not boot properly

I just found out my MacBook pro is eligible for the repair program but can I recover my data if it will no longer even boot up properly? It started with the display distorting as other users have described, but escalated within 48 hours to the blue screen and now will not boot up all the way. It gets stuck at the load screen with a distorted display and then shuts off.

I would buy a new computer, I'm not so worried about that, but I have files that have not yet been backed up that I really need to recover.

All advice is appreciated.