Can you boot an internal HDD with OS X as an external drive on another Macbook?
To boot from your HDD:

Get the hardware:


Get a SATA to USB adapter.


Remove the HDD from your 2011 MacBook.
Connect it the 2010 MacBook using the SATA adapter.
Power up the 2011 MacBook and hold the Option key until you see the boot menu appears.
Select your HDD (you will see its name) and boot from it.





To only access your data:

You can use Target Disk Mode (TDM):


Connect both MacBooks using the Firewire 800 ports.
Make sure the 2010 MacBook is running.
Power on your 2011 MacBook and press CMD+T to start it in TDM.
Your 2011 MacBook's HDD will appear in the 2010 MacBook's Finder


You should absolutely try to boot your 2011 MacBook's HDD using the TDM method.