To answer this, we need to explore the question of why you want a null hypothesis in the first place. What does that mean? What is its purpose.
A null hypothesis is typically what someone would expect if they were to believe the conventional wisdom of the times. In a scientific experiment, one designs the experiment to have the potential to reject the null hypothesis. Phrasing that differently, the experiment is structured to demonstrate a good reason to doubt the null hypothesis. If you were living in Galleleo's time, it would be conventional wisdom that a heavier ball will fall faster. One sets out to reject this hypothesis by running an experiment which yields data that suggests otherwise.
One can indeed make null hypotheses which are not aligned with the conventional wisdom of the times. One can make a "null hypothesis" that purple swans exist. And, if you gather data to refute this, anyone who believes there were purple swans now has some troublesome data which refutes their beliefs. But with very few people arguing for purple swans, such a hypothesis would not make very many waves in the scientific community.
The trick is to remember that the scientific method's construction of a null hypothesis and an alternate hypothesis is just formal rigor. In the strictest readings of the scientific method, one never actually accepts a theory. There's merely theories that have not been falsified yet. Thus, when one has done some work that makes one believe they have a new hypothesis, they don't get to prove that they are right. Instead, they must prove that the conventional wisdom is wrong, and after succeeding in doing so, they are given the privilege of suggesting what the next theory should be (the alternate hypothesis).
Now with that understanding of what the purpose of the null hypothesis is, we can see that it is reasonable to develop a null hypothesis to use with your claim "There are no purple swans." Now the "conventional wisdom" approach is tricky here, because you're really trying to make an experiment to show something that everyone already expects (which is rather boring). However, there's no reason we can't do it.
In the general pattern, we assume our claim is the alternate hypothesis. The null hypothesis must be something which contradictory to our alternate hypothesis. Typically this is easy, because we're trying to refute the status quo, so everyone will tell you what the null hypothesis should be. In this case, without a clear consensus that purple swans are a thing, it's a bit harder. We have options:
Alternate Hypothesis: There are no purple swans
Example options for the null hypothesis:
- There are purple swans
- One can assign the color purple to swans
- "Swan" is a word which is sufficient to specify an entity (which may have a color)
Practically speaking, the first one is the most natural. However, I wanted to point out others to show that all you need to do for a null hypothesis is to identify a hypothesis which can be refuted by data which is not your claim.
From that point, you could construct a test. Refuting "There are purple swans" is a tricky task, but it could be done with some reasonable level of rigor. An assay of populations in the wild which fails to demonstrate a single purple swan would be considered sufficient evidence in some fields.