Is this GCD statement true? If we have integers $h$, $i$, $j$, and $k$, would it be true to say that $\gcd(h,i)\gcd(j,k)|\gcd(hi,jk)$? If so, how can we prove it?
No, consider $h=i=2$ and $j=k=3$, then
$$\gcd(h,i)\gcd(j,k)=\gcd(2,2)\gcd(3,3)=2\cdot 3=6\nmid 1=\gcd(4,9).$$
This statement is false, You will see it whenever you take $h,i,j,k$ such that the products $hi$ and $jk$ are coprime but the numbers $h,i$ and $j,k$ are not pairwise coprime.
Illustrations:

Select $h=2, i=4$ and $j=3,k=9$, you will see that $6\not| 1$.
Select $h=5,i=15$ and $j=2,k=4$, you will again see that $10\not|1$.