I've download the HotMail App for my iPhone5 and iPad3 Both I type in my user name after hitting Login its sits and winds for minutes then dumps me back out to sign in screen. I have the SykDrive App for both as well and I have no issues.
Can't seem to find out anything at all from MS website about the program nor is it mentioned in any forums. I use IOS7 on both the iPhone and IPad. Gmailand Yahoo!Mail app work great.
I've IOS 7.2 the kind folks Apple have not seen fit to update the system to 7.2 yet, in the forums