Macbook Pro not Booting

I have 2 MBP that have both seemed to crash on me. One of them will boot to the Apple logo with a status bar across the screen. Once it gets about a quarter of the way across, the computer shuts down. I have tried to repair the HDD from the disk utility, but it shows it cannot be repaired. When I put the HDD in a cradle connected to my Mac Desktop, I am able to see the files, but it will not let me backup all of the files, only some of them. Also, when I look at the startup disk, there is not one to choose from.

On the other computer, it boots up to the folder with the question mark on the screen. I can boot to the Internet recovery and then to the disk utility, but have not tried to repair the files just yet because I wanted to get everything backed up first. When I cradle the HDD to my Mac Desktop, I am able to backup everything on the HDD. The startup disk menu looks the same as the other in that there is no option to choose from.

I'm posting to see what step I should take next. If I try to reinstall the OSX, will that delete any of the files on the computer? What are the odds that both of these HDD failed on the same day? I have a couple of friends who are having the same issue running Yosemite, which is making me think it's a software issue. But this is really frustrating because I haven't been able to find a solution if that is the case.