OK, I had a chance to examine the two files some more. The main problem is the wrapping of the h.264 video in the AVI container. I was able to copy the H.264 contents and rewrap to a mp4 container without recompression and I can import and edit them without the pixilation glitch. Here's what you need to do.
First, assuming you are on Windows, go to
http://ffmpeg.zeranoe.com/builds/ and download the current ffmpeg static build file. It's in 7-zip format, you may need to install 7-zip if you don't have something that can extract the files from that type of file.
Extract the ffmpeg.exe program from the downloaded file. It doesn't need to be installed, for convenience place in the folder containing the avi files. I'm going to assume that's what you did. It's a DOS program so you will need to open a command prompt to execute it.
Open a command prompt. Using dos commands, navigate to the location where the program and the files are located. When there, enter the following command line substituting the actual avi file name for "filename":
ffmpeg -i filename.avi -vcodec copy filename.mp4
For example, ffmpeg -i 20130505_092000_01.avi -vcodec copy 20130505_092000_01.mp4
This will extract the h.264 contents from the avi file and rewrap it into an mp4 file. Repeat the command for each of the avi files.
You now have valid mp4 files that you can assemble and edit. There is one last issue. The final frame in each segment may be bad. You'll need to check each. It looks like the program that split the clip didn't create a proper frame at the cut. Here's an example:
irfglitch.JPG
Since it's only a single frame, you could leave it and it's unlikely that anyone will notice. But, you may want to trim off the final frame. Not a biggie.
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