If you leave it plugged in and shut off or Reboot the Computer the External Drive is not getting Dismounted correctly and sooner rather than latter you'll discover that there is no longer anything on it. In a Worst Case Scenario the Read Write Heads will make contact with the Platters and totally destroy any Data on that Section of the Platter while mangling the Read Write heads. If you then continue to use the drive those Mangled Heads will then be dragged over the entire surface of the Platters inside the drive making any chance of a Successful Recovery [b]Very Unlikely.[/b]
You'll be prompted to Format the Drive and loose whatever you have on it. I very much doubt this is what you want to happen as it defeats the purpose of having an External Drive to begin with.
Before shutting down any computer [b]You Have to Dismount the External Drive[/b] by in the case of any form of Windows using the Safely Remove Option on the Task Bar. When you have Dismounted the External Drive you get prompted and you should then unplug it.
Then you can shut down or restart the computer without any problem or run the risk of loosing all of your Data on the External Drive. Remember that while a External Drive is relatively cheap the costs of a Professional Data Recovery on it will exceed thousands of $ so if you have any [b]Important Data[/b] involved on this drive you should look after it.
Also as External Drives where never designed for Full Time use leaving one plugged in when it's not needed is going to shorten it's life dramatically. Best Scenario is that the drive will live for 50% of the time that a Internal Drive would work for and if the drive dies it costs considerably More to recover Data off a Dead HDD than a working one. The last dead drive that i had to have data recovered from for a business cost $56,000.00 on a $80.00 to replace drive.
A new drive cost $1.00 a Gig to buy back then and it's considerably cheaper now but cost $700.00 a GIG to recover. While the cost of a GIG of HDD space has dropped the cost per GIG of HDD Space in Recovering Data has not changed so consider the costs involved in Recovering Data off this drive when it stops being recognized or working. ;)
The other possibility involved here is a Voltage Issue if this is a 2.5 Inch USB Drive it gets all of it's power through a USB Port and this could be pulling the 5 V DC Rails in your Computer Low preventing it from starting.
This would indicate a Marginal Power Supply in a Desktop Computer or a Portable Computer in use with an even more marginal Poorly Designed Power Supply System.
Col