Output voltage of AT28C64B into HC CMOS IC
Few different solutions. In general TTL outputs are made compatible with CMOS inputs by using pull-up resistors. Put pull-up resistors on EEPROM output to make them go up to 5V, and switch the shift register to 5V. Another solution might be to use HCT type shift register that has TTL compatible inputs, but it would still require 5V supply for the shift register.