are 100K resistors too weak for connecting HC to cmos?
the pull-ups I described above are connected to the outputs of the
  shift registers (74HC164)... is 100K too high of a pull-up resistor
  value between an 74HC output and a cmos input? If so, what is the
  maximum value I should use?


The 74HC164 already has CMOS outputs that actively pull up to Vcc, so you shouldn't need any pull-up resistors.


  The output of the registers are connected to a AT28C256 EEPROM


The AT28C256 has TTL compatible inputs, so pull-ups should not be required even when driven with standard TTL logic.


  I was told that a solution was to use the 74HCT164 instead of 74HC164


This sounds like there might be a problem with voltage levels on the inputs to the shift register. HCT logic has TTL compatible inputs (0.8V-2.0V), whereas HC requires CMOS input levels (1.3V-3.7V). If your microcontroller's I/O is 3.3V it might not work with standard CMOS, but should work with HCT.  

The chart below shows input and output voltage ranges for various logic families. For an input to be reliably recognized as high or low it needs to be outside the purple/blue areas. Outputs normally stay inside the orange or yellow area. 

Standard CMOS, HC, AC and AHC all have the same input and output levels. Their outputs are also compatible with HCT and standard TTL.

Standard TTL outputs may not be compatible with standard CMOS inputs because a TTL output can be as low as 3.3V. This is where a pull-up resistor helps, since it can pull the TTL output up closer to 5V.