It doesn't sounds as if the real time requirements are particularly tight here since they are basically to do something a couple of times a day for a few days - I think the Arduino's own timing would give you enough accuracy for what you need without the complexity of an external RTC.
You need to decide whether you need any feedback about what it's doing (how far through the program is it, what is the actual temperature - that sort of thing). If you can leave it attached to a PC then you could just print status messages on the serial port and do without any separate display.
It seems to me that the hardest requirements here are the mechanism to turn the egg (the solution to that will determine whether you need a servo or a stepper motor - they are both easy to use) and some method to read and control the humidity.
If this sequence can be predefined and just start when you power up the Arduino, you don't need any controls. But if you may need to reset the Arduino and resume the sequence then you would need some way to tell it where to resume the sequence from. Again, if you can afford to leave it attached to a PC you could do that by sending commands over the serial connection, otherwise you would need to give your Arduino a display and some buttons to tell it what to do. The software to drive the display and buttons is quite simple, but all adds to the complexity and if you're in a hurry you'd do well to avoid them.