Performing API calls from a mobile client, reauthorizing the session if necessary
I agree with Ibalazscs that StatePattern makes sense - using a switch to drive changes in behavior is a code smell.

Given that this state machine has only one end state, you could ditch the processConnect boolean, and just check for end state...

ConnectionState state = ConnectionState.checkAnswer;
while(! ConnectionState.endProcess.equals(state) {
  ...
}


You also might want to consider whether ConnectionState should implement an interface to make it extensible.  See Item #34 "Emulate extensible enums with interfaces", Effective Java Second Edition, by Joshua Bloch