Can I use an ATtiny84 as SPI slave and i2c master in the same application
Is the best solution to use a tristate buffer that my ATtiny84a can toggle to cut off the external SPI master from the lines in question whenever it wants to read/write to the EEPROM?


There may be a simpler solution.


  ... ATtiny84A which uses a USI (Universal Serial Interface), and as such there is no dedicated I2C and SPI peripheral.


Well, USI is a dedicated I2C and SPI peripheral.  But only one kind of bus in any given design.


  Can I use an ATtiny84 as SPI slave and I2C master in the same application


Yes, that's possible.  


Implement the I2C bus master using software bit-banging.  This allows you to use any GPIO pins for the I2C bus.
The I2C buses usually run a leisurely rate, and bit-banging an I2C master in firmware is a viable option.  
Use the USI hardware for SPI slave.
Implementing an SPI or I2C slave in firmware is impractical at any reasonable data rate.