This claim is based on the information collected by the source provided, and is not a researched or truthful theory. A German scientist Tributsch published a book regarding a theory of ions rubbing together and animal skin being a radar, or detector, of natural disasters such as earthquakes. There is no mention of research, but rather just a mention of this estranged theory.
Based on personal research and internet surfing, there has been no set up trial or research conducted where animals, specifically dogs, are known to predict earthquakes. Articles such as, Can Dogs Predict Earthquakes , go into the depth of the characteristics and trigger warnings of earthquakes, and assumptions based on the science and composition of dogs, claiming them to be quake predictors. The date and information of this source shows how this issue is a hot topic, but in short doesn't do much more than what the original source of this claim did with its message delivery.
For cats, according to an article published on ISAAC, it seems to hold the same provided above rationale. There is a hypothesis for cats having a special sensitivity to the earth's magnetic fields, which aren't perceptible to humans, based on the theory of Tributsch.
There is not much out there besides theories, and hypothesis based on Tributschs' work, thereby falsifying this claim.