So the Sony HDR-HC1 has a microphone input that is a stereo mini plug. So what I'm wondering is, could I plug in a "Sereo Mini Plug to XLR" cable and get high quality sound by plugging in a Sennheiser ME66 to the XLR? Will this be as high of quality sound as it would be if the XLR plug was already on the camcorder? I know the Sony HVR-A1 has 2 XLR plugs already on the camcorder, but it is more expensive. So I was just wondering if this would work.

Something like these.

http://www.equipmentemporium.com/videoaudio.htm

I like the audio boxes on the top of that page, but can those be used with the Sony HDR-HC1?

Yes, you can plug the ME66 into the HC1. It might be a little high in output, so you'll want to be careful. The Beachtek boxes will work wonderfully with the HC1, yes.
XLR vs 3.5mm makes no difference to sound quality. It only makes a difference as to how long a cable you can run might be.

Bear in mind, the HC1 has a stereo mic; plugging in the ME66 will give you a mono recording.

I guess it would be good if it was mono. In my opinion, I think it would be better to have mono if recording dialouge in movies, so if I mix the audio to be 5.1 Surround Sound, I could place the dialouge where ever I wanted.

You virtually always want audio in mono when recording dialog. You also might want to spend a little time in the "Now Hear This" forum on this site, you'll find loads of great audio advice.