Shorting ground and left (or right) of 3.5mm audio jack from the computer
Pretty much all Canon EOS SLR cameras are able to use the Canon EOS software for semi or complete control from a computer. Heck, you can use your computer as a view screen as well.

Aside from that, a USB GPIO or Relay board will do what you want. You may have to add two transistors for the GPIO board, but that is pretty simple.

There is no standard way of using a headphone jack to short a connection, but there are some hacks that could do so, and some commercial products.

ioShutter is a cable + app combo for iPhones. TriggerTrap and TriggerHappy are similar for Iphones and Android.

DSLR Remote is a free app for Android with both commercial and home made cable support. Two of the cables are simple, with transistors or optocouplers
As long as the headphone out's maximum volume/output/voltage is high enough, it should trigger the remote. The audio source files arn't included, so the software side you would need to make yourself.

This site shows a serial port method for windows, with software. While this site expands on it, with both shutter and focus, with linux software. You could even use a Usb to Serial adaptor with these.