Arduino due dying after connection, gets hot
The "504L" is a 500mA polyfuse.  If more than 500mA is attempted to be drawn through the USB port it will get hot which causes the fuse to "trip".  When it cools down again it will "reset".

Why is too much current being drawn through the USB port?  Well, that's what needs answering to know just why your board is misbehaving:


Do you have lots of things plugged into the board - like LED strips, motors, etc?
Is the board sat on something metal causing a short circuit? (I did that by having some loose change on my desk once).
Have you physically damaged the board and caused some pins of one of the chips to short together?
Has the Magic Smoke™ escaped from one of the chips rendering your board broken?