Some times it's a good thing to avoid the /illusion of evil/. You might not be doing anything bad, but it would sure look like it.
If you really want to know if their system's online or not, convince them to do instant messaging. Have them send you a message when they've got it up and running. You can also do some remote support type things (I find users tend to like things written down anyway) with that.
Old computers are getting to be a lost art. Here at Uncreative Labs, we
Dunno where you are, but there are laws on the books in the USA (and elsewhere) regarding unauthorized access via network.
One case which received national attention comes to mind. A technician repaired a client's (in this case corporate) network, then proceeded to remotely run a server check. He got in hot water for using his client's network in a manner not specifically detailed to and agreed to by the client, even though he was contracted to perform repairs on that specific piece of their equipment, and that server test was a necessary part of the repair!
Moral of the story: If you are intent on doing this, spell out to your customers EXACTLY what you intend to do, including the software you will use, the time at which you will have their system send you info, the exact info their system will send you, the length of time your software will be used (and installed) on their machine, the effects, if any, your software will have on their security programs, etc. etc. Maybe they will agree and maybe they won't appreciate it AT ALL. I think you'll find the latter to be the case. Do it covertly and you open yourself up to prosecution if these laws apply to your area, or potential litigation even where no such criminal laws exist.
Unless you absolutely must interrogate their system in realtime, just have them send you a courtesy email. Or make it mandatory if you think they won't hate that (and they will) -- state on the receipt that if they elect not to do this, you won't accept the return if there's a problem. It's their choice.
An email will tell you a lot about their system, such as: They are connecting with their ISP, which speaks to their modem function, IP stack health, DNS resolution, etc. etc. Their email client is working, which is a good indicator the OS is largely if not fully functional, particularly in the case of an integrated client like OE.
I would definitely choose a less invasive course of action than what you propose. Users are scared enough about spyware. They would certainly ensure you get a bad reputation were you to give them additional cause for concern.