No, that analogy can't be applied in reverse. You're right that placing the burden of proof on the positive assertion is a dialectic oversimplification. But the specific claim, "A living creature cannot be as big as a galaxy" does not suffer from the problem of induction - we can use deductive reasoning to falsify it given relevant, empirical observations.
The classic example of white swans and black swans adheres to the problem of making universal generalizations because we simply have not yet observed a black swan, and we cannot use the continued observation of white swans as empirical evidence. However, we are not attempting to make a universal generalization based on inductive reasoning.
We have, available to us, a schema for enforcing the testability of this claim, which the article and its references explore in detail. The square-cube law and our current model of physics both make it impossible for a "living creature" to be as large as a galaxy, for any definition of "living" and "creature" that is reasonably agreeable.
The normal processes of life; including temperature regulation, metabolism, energy expenditure and movement would all be severely impeded in a single discrete organism of that size. Moreover, it would have severely limited cognition even if signals traveled at the speed of light, which handicaps both its own sophistication and potential evolution.
It doesn't seem appropriate to characterize something that cannot have these functioning traits as "living"; the only way forward seems to be to broaden our definition of "living" or "creature" to some sort of aggregated, abstract consciousness that makes organisms like ourselves analogous to bacteria. But that can hardly be called a discrete organism in the same way that our brains are cognitively independent of the life around them.