Let's suppose that we're summing $m$ and $n$ many odd numbers starting from $1$ separately, i.e
$$S_m = 1+3+\cdots +(2m-1)$$
$$S_n = 1+3+\cdots +(2n-1)$$
What can their differences be among $37, 42$ and $60$?
I started off with $S_m = m^2$ and $S_n = n^2$, so their difference becomes
$$|S_{m}-S_n| = |m^2-n^2| = |m+n||m-n|$$
But I am not sure what else can be done outside of checking particular values of $n, m$.
You have correctly shown that the set of possible differences between $S_m$ and $S_n$ is the same as the set of positive integers that can be expressed as the difference of two squares. From Proving expressibility of integers as the difference of two squares., all positive integers can be so expressed, except those that are $2 \pmod{4}$, so in particular, $37$ and $60$ are possible, but $42$ is not.