The short answer is yes, it certainly did happen. The cat (and pets in general) was sometimes seen as a symbol of bourgeois decadence and practice that needed to be uprooted from Chinese society. I base this conclusion on Dikotter's A People's History of the Cultural Revolution and the following passage:

Dogs had long since disappeared, denounced as a threat to public hygiene before they were hunted down in cities across the country in the early 1950s. Now came a great cat massacre, as Red Guards did the rounds to eliminate the feline symbol of bourgeois decadence. Rae Yang, the fifteen-year-old student who had turned against her teacher in Beijing, tried to smuggle her pet out of the house, but Red Guards noticed something moving in the bag she was carrying and guessed what was hidden inside. They grabbed the bag, swung it round and hit it against a brick wall. The cat mewed wildly. ‘The boys laughed. It was fun. They continued to hit him against the wall.’ Her brother started to cry and begged them to stop but nobody listened. Walking through the streets of the capital at the end of August, people saw dead cats lying by the roadside with their front paws tied together.

Chapter 7, pg 109

That being said, I have great reservations about the implied scope of both the crackdown on dogs and cats in the post 1949 era. His citation for this account is Helen Wang's Chairman Mao Badges: Symbols and Slogans of the Cultural Revolution. Nothing wrong with this source, but it does not actually conclude that there was a widespread campaign against cats and dogs for a prolonged period of time. The Cultural Revolution was marked by rapidly changing standards of what was and was not acceptable. These new norms quickly came and went as different factions and groups sought to assert their revolutionary credentials by demonstrating the proper zeal. As such, it seems likely that this "cat massacre" was much smaller than Dikotter implies and limited only to Beijing for a short while. In the smaller towns and villages, both dogs and cats were useful animals to aid in agricultural production and would not have likely been seen as symbols of a bourgeois lifestyle. Much of the zeal associated with the cultural revolution was associated with the cities. So much so, that many accounts of red guards going out into the countryside note the cultural divide between urban and rural Chinese at the time.

If there is a better source out there that better details this phenomenon, I would greatly appreciate another member of this subreddit to let me know. Thank you.