My external hard drive will not stay connected  

that is really strange. i bought one of them for my mother in law and it works great. the only time the lights go on and off is when it is working. you should call customer support. a lot of companies will let you send it in for repairs or they will walk you through how to fix it over the phone. hope this helps

Have you check your connector and your port (on external drive and on your PC)? From what I got here I think the problem is somewhat on your port/connector.

Try to attach other device to that port using the same connector, then try using another port and another connector on your external drive. Experiment with it and then you will be able to identify what's wrong with your device.

Go into my computer, right click on your external hard drive --> properties, then there is something like disk utility. click that and do disk repair on your external hard drive, I will explain better when I am on my windows computer.

Its the problem of the USB Port or the cable that you use to connect the hard drive to the PC

How about this one,

First try to check your power supply, do you hear a noise in your power supply?
Try to install your power supply into your friends cpu, if the same thing happens to your friends cpu then that should be it, may be its already old or watts is not hi enough to support all of it. Check it out first, if its not. that would be another case.^^