The search field does not expand when you click the search button in the everyone screen on the iOS device, and this is a known issue in the Mobile Time Tracker application.
BTW, You can search for a user by clicking the search button and entering the name that you wish to search. As the search field is not expanded, and you cannot see the name that you entered in the search field. But, after you key in the name and click the search button (on the keypad) you see only the user name that you search for with the Clock-In/Out or the Check-In/Out buttons.
We will fix the issue and release it in the future release of the Mobile Time Tracker application.