First of all, very shortly: A video file consists of a stream with a codec (which is the method the video and audio is compressed) and a container (which "contains" the stream and some information like resolution, framerate, ...).
The file endings (.TS, .MP4 or .MOV) only tell us the container format that you use, and nothing about the encoding of the content. The easiest way to find out the codec is to click on a file and press "Command+I" for the information window. This will however only work for containers that Mac OS knows well, for example .MOV or .MP4.
Now, I have two suggestions for your problem:
1. If it really is just about the video container (I thought FCPX could handle .MP4 files, not sure about the others though...), you can try converting the video files with
this little tool I wrote into a .mov container and see if FCPX will import this. Out of the choices you wrote, I would definitely go with .MP4 by the way, if all three record the same codec.
2. Try to find out the codec of your friend's videos. If Command+I reveals nothing for the .MP4 file, he can try to open a video file in
VLC and press "Command+I". In the window that opens, go to the "Codec Details" section and flip down the stream information. Tell us what the codec is. Although, in my experience, FCPX will be able to deal with a lot of codecs.
I hope this will help a little. I am very surprised though that the .MP4 file wouldn't import into FCPX. Do you get any error message? And what does it say? It's kind of hard to tell with so little information what could be going wrong.