I don't think RFID would work here; it works for a cat because a cat holds its head low and is predictable as to where the dangling RFID tag will be when it walks up to the door. The RFID reader you have linked (well, the non-discontinued version) states a 7cm sensing distance. I have no idea how you'd leash an RFID tag onto a chicken to get a predictable 7cm position and reading.
I'd suggest a simpler method would be to use some sort of light beam across the nesting area inside the nesting box; when the chicken blocks the beam it's in the box. An infrared LED and infrared phototransistor can be wired up to do this.
You might want to look into the Seeeduino Stalker which includes an SD card socket and real time clock right on the board ($39). It also provides a solar panel input and LiPo battery input so, if you keep your current consumption low, you might not even need to run power out to your house.