On my Windows or Mac computer, I turned off my Bluetooth receiver, or unplugged my Bluetooth dongle from my computer, and it shows a message that the “ThinkGear Connector has crashed”. Did I break it?

There should be nothing wrong with your computer. There is a known issue where the ThinkGear Connector service simply gets confused by the sudden abrupt removal of a Bluetooth device. If this happens, simply restart ThinkGear Connector from your Windows Start Menu or App Central, or from your Mac’s Applications folder, and it will be working properly again.