This is not a repair I would recommend trying for someone who is unfamiliar with camera repair. There's a lot of delicate work involved in getting the proper tools in and out of position to remove the focus screen in the 600D without damaging the mirror or external side of the focusing screen, both of which are very easy to scratch and damage. And anything that changes the reflectiveness of the mirror or the transmissive properties of the focusing screen can also affect the accuracy of your camera's light meter, which is located above the focusing screen. The light measured by the meter must first reflect off the mirror and pass through the focusing screen.
Any good camera repair facility can remove, clean, and reinstall the focusing screen in a few minutes. If you live near a large city that still has a larger camera store they can probably do it for you at a relatively inexpensive price.
Sending it off to a repair center gets expensive very quickly just from the standpoint of shipping and insurance. It never hurts, though, to call the nearest Canon service center and get an estimate. Also check with your courier of choice to see how much it will cost to ship and insure your camera to the service center.
Although Canon does not officially consider the focus screen of the Rebel T3i/600D user replaceable, the method to remove/replace the focusing screen is very similar to models that are considered as such by Canon. If you choose to try it yourself please do so at your own risk! You must accept full responsibility for whatever happens to your camera. We accept no liability for any damage resulting from your choice to use the instructions linked below. Should damage occur as a result of following these instructions you accept full responsibility for the cost of any servicing, repair, or correction needed. Here are the instructions from katzeye optics for removing/replacing the focusing screen of the 600D. Please also be aware that even after successfully removing the focusing screen, it may or may not be possible to clean the residue of the dead bug from the screen without damaging the delicate surface of the screen.
Some Canon camera models do have focusing screens officially considered user replaceable while others do not. Many of the higher end models that do not support user replaceable focusing screens include LCD overlays for displaying information in the viewfinder. The early models with the LCD overlays (7D, 5D Mark III, 70D) do not support user replaceable focusing screens. Due to the presence of the LCD overlay the removal/replacement of a focusing screen on those models is even more involved and delicate that that of your 600D. The newer 7D Mark II and 6D include an LCD overlay but also have focusing screens considered user replaceable and several alternate focusing screen options are available from Canon. None of the Rebel/xx0D/xxx0D series or the x0D series prior to the 50D officially support user replaceable screens either. Some of the newer models in those series may also include an LCD overlay.
Here's a list from Canon as of 11/04/2015 that includes which cameras have officially user replaceable focusing screens and which replacement screens are compatible with which models. (The newest Canon models, the EOS 5Ds and EOS 5Ds R, are not listed on the 11/04/2015 version of the list)