There is a trap that you might not have thought of.
With least squares it is possible that the simple answer gives you a theta which is can't be true. For instance if the X's are 10, 10, 10, and 20, then the average is 12.5 and theta would be 10 + 2x2.5 = 15. But 20 is impossible in U(10, 15)
Interestingly, maximum likelihood gives a different answer altogether from least squares.