MacBook Pro 13" Late 2011 - shuts down during white Apple screen

That signals defective or unseated RAM modules. I suggest you do the following:

1. Shut down the computer, remove the AC charger, and remove or disconnect the battery.
2. Be sure to discharge any static electricity from your body by touching the computer's metal case.
3. Access the RAM slots on the computer and remove the RAM modules.
4. Gently clean the slots of any dust or debris.
5. Get a clean pencil eraser and use it to clean the edge card pins of the RAM modules.
6. Reseat the RAM. Check that they are properly seated in the slots.
7. Reconnect the battery.
8. Connect the AC charger.
9. Reboot the computer and see if you still get the three beeps during startup.

If the beeping continues, then you have one or two defective RAM modules. Usually, you can remove one at a time to check which one causes beeps. Replace the defective module(s). Or, just buy two new modules and replace the old ones.

In addition to @Kappy's excellent advice I would like to add this little bit of information. A lot of times the lower memory slot nearest the Bottom Case will go bad due to broken solder joints. Try removing just that module and testing the memory in the slot nearest the Keyboard. If you still have problems, then test the memory in only the slot nearest the Bottom Case.

Nice addition, HW. Thanks.