Oh, interesting. They even made a movie.

https://en.wikipedia.org/wiki/US_Airways_Flight_1549

Once an engine breaks the question is whether or not it becomes unbalanced, that is one reason why they can become detached from the plane.

The account on that page seems to indicate that the engines didn't detach as a result of the bird strike. One detached upon impact with the river, and the other was still attached.

Unexpected way to increase air safety:

> In an effort to prevent similar accidents, officials captured and exterminated 1,235 Canada geese at 17 locations across New York City in mid-2009 and coated 1,739 goose eggs with oil to smother the developing goslings.[71] As of 2017, 70,000 birds had been intentionally killed in New York City through programs instituted after the ditching.[72]