The Arduino EEPROM library is compatible with the ATTiny range of AVR microcontrollers as the library itself is built on the standard Atmel AVR avr/eeprom.h 'library' so it is compatible with all the AVR microcontrollers.
The EEPROM also doesn't take to being written or read to often as EEPROM can wear very quickly. Reading though does not cause much damage though.
Also note that using exit(0); will stop the ATTiny from doing anything else after it us called so I hope your intention is to only run the loop once.
To answer your follow up question. Yes you can run one sketch to set you values in EEPROM and then use another sketch to read those. That us usually the point of EEPROM it is a memory type that "keeps its value while there is no power".
Also you need to make sure the ATTiny is set to preserve EEPROM during upload with the ISP, this is done with the fuse settings. You need to look for a tutorial on fuse calculators for the AVRs. I can't recall what the settings would be off hand but will add them later.