Is it true that if any two of $m$, $n$, and $mn$ are sums of two integer squares, then so is the third?

Question

Where m and n are positive integers. Prove or give a counter example.

Answer 1

The following identity, often called the Brahmagupta Identity, will help you: $$(a^2+b^2)(x^2+y^2)=(ax\pm by)^2 +(ay\mp bx)^2.$$

Answer 2

A non-trivial characterization is:

An integer is a sum of two squares iff its prime factors of the form $4k+3$ appear with even exponent in the factorization.

This will make it easy to answer your original question.