ah no sorry, I made myself unclear, the character controller itself is a collider already.
About the transitions :
each state can have multiple transition and multiple actions. Eg. you could add several Collision Events, one which triggers if a mesh enters that has the tag "untagged", and sends the event/triggers the transition "blueberries", and another collision event which triggers if a mesh with the tag "player" exits and which triggers the event "pancake".
This means an untagged mesh has completely different results than a player mesh, because the 'blueberry' transition might lead to State 1 and the pancake transition would lead to State 2.
Since the state machine works (just like in hard code) from top to bottom of a state, you could also create a lot of actions that might not happen at the top of the state, like gameObject Compare and then try to see if it is the game object you're looking for. If you do only specify an event in the "Is True" property of that action, no transition will be triggered if the gameObject is not the one, and the stateMachine will start looking at the next action in the state. You could repeat that for each gameObject you think should be needed and then finish the state with a "send event" action, which always sends an event, no matter what, but since it's at the bottom of the state this action will only trigger an event if no other action did so before.
So about the setup, what disastercake wrote should work, I'll just describe it more thoroughly:
1) check if both objects have a collider or character controller on them. The mesh with the FSM on it should in this case be a collider and the other one should be a character controller or a collider + rigidbody.
2) on the egg (or whichever object should recognize the collision), set the collider to Is Trigger.
3) create an fsm and right click at the start state. Create a transition. Call it FINISHED (always handy to use the default ones, so it won't clutter you up)
4) add another state and connect FINISHED to that state
5) in your start state, create a Trigger Event
6) make sure the tag matches your other object's tag
7) set Send Event to FINISHED
run it