Will this reduction of Exact Cover into Subset-Sum fail due to a potential false positive?
If you're not really sure whether a reduction will work, it probably won't. Whenever you are making a reduction you should always have a plan of how to prove it correct.

In this case, we're looking to see whether $1^2+2^2+3^2+...+n^2$ can be written as a sum of squares in some other way (which would be a false positive).

We have that $4^2=2^2+2^2+2^2+2^2$. This is enough to construct a counterexample.

Let $S=\{1,2,3,4,5,6,7\}$ and $C=\{\{1,2\},\{2,3\},\{2,5\},\{2,6\}\,\{2,7\},\{1,3,4,5,6,7\}\}$.

There is no exact cover (the only way to get $4$ is to take $\{1,3,4,5,6,7\}$ but then we cannot take any other set since all of them overlap).

However, we have that:

$$1^2+2^2+3^2+4^2+5^2+6^2+7^2=(1^2+2^2)+(2^2+3^2)+(2^2+5^2)+(2^2+6^2)+(2^2+7^2)$$

Thus, we can conclude the reduction does not work.