I'm pleased to report that I have had no issues in the last week. However, two changes have taken place:
1) My client gave me Cisco's latest AnyConnect VPN to install on my Windows 7 Ultimate PC. In fact, I believe I am the beta for this new release.
2) I became convinced that it was indeed the VPN that was causing Windows 7 to crash. Therefore, I took it down whenever I didn't need it rather than just leaving it connected all day.
So far, so good.
In answer to your questions, I'm running a wireless connection with a cable modem and a LinkSys 24 GHZ router. I do not use a wired connection any longer. There is only a single PC involved in this issue. It is a new HP Touchsmart 600 with 8 GB RAM; a very nice machine, notwithstanding the blue screen issues.