You are done.

If you have correctly found the joint density of Z1 and Z2

and it factors into two functions, g(Z1) and h(Z2), which are NOT necessarily the marginals,

AND the support is a rectangle, then the two rvs are independent. Most books leave out that fact.

BUT, you can get the marginal densities of Z1 and Z2 and show that their product gives you the joint density, then you have certainly proved that Z1 and Z2 are independent.