Ahhh, sounds like you have one of two or three things, first there is either spyware, notorious for shutdowns, second an overheating issue, you should get a warning, but not always. second , windows is corrupt, and the kernel reboots, does it do an NT/Authority Shutdown? or just gone, if its just gone that could be the kernel reboot. You can repair windows, either reinstall , roll back, (not really a fix cause the problem could be way back. You should go to the administration section in your windows and look at athe event log, it should say why or give some shutdown errors. that will lead you to the problem cause. The guy on the phone is just reading out of a book, he has no idea what or hw to fix anything. Ask him to transfer you to the second lvl support, he'll fight you but push for it, or demand a service call. The first thing they will do is replace the motherboard, and/or check your memory. I would reccomend that you remove all but one memory stick from the system and see if that helps, if not change that memory for the other stick, assuming you have more than one, memory can do that as well. In a few changes you should find the bad stick if that is the problem, if it isn't that they will replace the sytem board, but on your end you should run an updated virus scan, and a spyware cleaner as well, I reccomend either spybot or adaware, both are free. delete all that they find, that you can do at home. See if they will send a tech, yes its a pita, but he will come with memory(should) a motherboard, powersupply, thats all it could be, now again in your bios, (usually the delete key) you can sometimes see a run DMT test you can run that and test the hardrive. But again if you are warranty play the game let them help you. Its what you pay for when you purchase there computer. There is more you can do at your end, go to start/run/ cmd when your black screen comes up type in after the c: prompt "sfc /scannow" don't use the quotes, it might ask for your windows cd. if so put in the cd drive and hold the shift key so it won't start.Then let SFC/scannow run it won't hurt anything, if there are windows errors it should fix them, and you won't loose data.
Second thing, when above is done, you may have to reboot, so after that go back into the cmd, window, this time type in chkdsk /r this will probly tell you it wont lock the drive becuse its in use, or ask you to force a dsimount, you can say yes, but it probly won't do it, but it will ask you if you want to do it on the next reboot , say yes and reboot. again it will inspect the drive for errors, repairing if possible, it will also check the system files again from a different view. Those are a few things you can try on your own, If you have any questions feel free to ask. And if your really desperate pm me. we can work over the phone or chat.
Support is my world.
If he is in chat with Dell he most likely has a warranty? That should include a tech on site to repair it.
If you ask for help you can't tie the hands of the guy thats gonna do it. They won't give you a free one just because a user won't let the guy access the computer. I suppose the laying of the hands could work, but only if you have faith! Every tech in our org. is security cleared, background checked, if you can't trust the computer tech, you aren't going to get a working computer for long. Dell, like all computer company's has to have there techs backgrounds checked. I personally get a LIEN check every 6 months, and updated background checks, yearly. If you have stuff on your C: drive that you are afraid of people seeing, it's your own fault if you loose it, you can be hacked so easily its almost a joke. The only reason you don't get hacked is that you don't have what they want, that and the sheer numbers of computers on the internet. Like I posted here yesterday, the worlds largest credit card company/debit cards was hacked. They have in my opinion a relative high security system in place. And its hacked yearly, 2008 being the largest in history. Backup your stuff, keep it on a separate hardrive and disconnect it from the system unless your doing an upgrade or back up. That keeps the system secure and you don't have to worry about someone going thru your personal stuff.
It also allows the tech guy to fix your system without you worrying about his looking thru your system, Which techs really don't care whats there, we see it all, its hard to surprise the techguy.