Callback on Bean save? Is there any callback hook like hook_node_insert / hook_node_update when saving a bean block?
Beans use the entity system, so you have a couple options here:

As AyeshK stated, hook_entity_insert and hook_entity_update are the obvious options.

If you need to do something before creating OR updating you can use hook_entity_presave.

Finally, Drupal doesn't have any native hooks that runs after an entity is created/updated, but the Hook Post Action module offers a nice set of additional functions to solve this problem.  If you don't want yet another module dependency, read the module's codebase and figure out how it implements the post-save functionality, then add similar custom code in your own module.