I would suggest storing the data in local storage with the following commands.
Set the data in localStorage can be done with the command:
localStorage.setItem('nameForData', variableNameForData);
To retrieve the data when required.
var variableNameForData = localStorage.getItem('nameForData')
To remove the data from localStorage:
localStorage.removeItem('nameForData')
These would typically be put inside action creators with a dispatch to change the state of some Boolean that tracks the applications interaction with localStorage.
For example you might have a state that is set to true when the local storage is created.
On refresh you might call an action creator that checks the local storage exists, and if it does set that boolean back to true or if it does not exist you are back to creating local storage and then set it to true.
You could put this function in componentWillMount(){}
and it will be called when the component is first rendered, and thus in the case of a refresh.
Docs for component life cycle and specifically componentWillMount here
Docs for local storage here