Health is a deal breaker for me. If it is lower than about 70 I usually look for a way to deal a position player, regardless of his other ratings.
Durability is a close second for position players. Of the ratings that actually help you on the field, I don't like Eye to be low.
For pitchers, Control is probably my biggest deal breaker, anything below 65 and they better be darn good otherwise. I cannot remember ever having a pitcher with a Control below 50.