What woods can be stabilized? Are there any that should be avoided?

Most any wood or other porous material can be stabilized.  You will have to decided if really hard or dense woods will give you any benefit by being stabilized.  If you are knife maker or call maker, stabilizing dense woods may make sense because the stabilizing will help reduce wood movement due to moisture changes.  For pen makers or other who can put a CA finish on their completed items, thus sealing them against moisture absorption, it may not make a lot of sense to stabilize those hard woods. 

It is best to not mess with trying to stabilize oily woods.  On some woods with high oil content, the vacuum process can actually pull the oil out of the wood and then it can mix with the Cactus Juice.  If the concentration is high enough, it can affect the chemistry of the Juice and cause cure issues where it may not cure properly.