Is there a chain reaction after protein starts to crystalize?

Not strictly speaking. Chain reactions produce unstable products which then go on to trigger further reactions. Protein crystallization produces no unstable products. As in other kinds of crystallization, once the process has begun through nucleation, there is an ever-increasing pace to the incorporation of adjacent molecules. But the nature of the bonding (hydrogen binds, salt bridges, and hydrophobic interactions) is such that there are no by-products.

At the risk of being hopelessly pedantic, I think we need to pin down what is meant by “chain reaction.” Dictionary.com gives:
— n
1. a process in which a neutron colliding with an atomic nucleus causes fission and the ejection of one or more other neutrons, which induce other nuclei to split
2. a chemical reaction in which the product of one step is a reactant in the following step
3. a series of rapidly occurring events, each of which precipitates the next
—————

So I think sense #3 applies to rapid crystallization once nucleation occurs. I recently saw a tv documentary showing supercooled liquid water in a glass jar suddenly and rapidly freeze once seeded with ice. I think the “ever-increasing pace” described by @thorninmud qualifies as a chain reaction in the most general sense of self-reinforcing. Sorry I don’t know anything about crystallization of proteins.