I just bought the Panasonic SDR-H40 camcorder and I have a problem with the image. When I film in 16:9 (Widescreen) and transfer it to VideoCam Suite (the program that came with the camcorder) on my computer, the image is as it should be. But once I copy it to my computer's harddrive and want to play it in another player (RealPlayer or Windows Media Player) the image gets squeezed horizontally to a 4:3 size.

Why does it do that and how do I fix it?

The video camera records the video as a 720x480 resolution file, but doesn't provide proper information in the file for media player software to know that 720x480 should be played as 16:9, since it's typically a 4:3 display format. Your best bet would be to open the camcorder file in a video editor that supports 16x9 video and save the file as a WMV, AVI or some other file format that will then be properly recognized as 16:9. You might also be able to play it with proper aspect ratio using VLC Player

I downloaded VideoCamSuite and, indeed, the 16:9 videos appear in their proper aspect ratio. Now, maybe it's just me, but I can't seem to find a way to save the videos.

Is there a player out there that has the ability to save the video? Or does VideoCamSuite have that option and I'm simply blind?

You need to use some type of video editing application to modify and save your videos. A player is typically only that - a player.