The EEPROM library works in Galileo, however it’s not the same library as the one used with the Arduino boards. The EEPROM library that is compatible with Galileo only has two methods:
And there are only three examples that you can run:
The original EEPROM library is used to manage the internal EEPROM memory of the ATmega used in the Arduino boards. Galileo doesn't have that EEPROM memory so the original library is not compatible with Galileo. The version of the EEPROM library that is compatible with Galileo only includes the two main functions used with memories: read and write. That's why you got that error, because you are trying to use the put() method which is not available in the EEPROM library for Galileo.
Unfortunately there isn't another EEPROM library for Galileo that includes more methods.