Solution:

Suppose we have $x=2+k$ as a solution. Then, $x=2-k$ would also be a solution as $$((2+k)-2)^2 = ((2-k)-2)^2$$ The sum of these two solutions would be $4.$ Thus, the sum of all integer solutions to the above equation is four times the number of positive $k$'s that work.

To find the number of $k$'s, we need to find the number of positive solutions to: $$1 < k^2 < 25,$$ which would be $3,$ as $k=2,3,4.$

Therefore, there are a total of $4\cdot 3=12$ solutions.

Thus, the correct answer is B .