Can you request a new squawk code if you don’t like the one being assigned?
No, there is no procedure for assigning a new squawk code because a pilot is superstitious. The whole system behind assigning squawk codes is fairly complex, and requesting new codes for non-operational reasons would put extra load on the system for no reason. In many cases, the controller doesn't even have the option to manually change the code, and doing so could result in significantly increased workload (actually getting the code, updating the flight data record, communicationg with the pilot, coordinating with downstream sectors etc.).
Squawk codes are a limited resources, we only have 4096 of them, and the system is already near its limit in busy airspaces. If we started to change codes just because we feel like it, we may very well run out of codes. And how should we choose which codes to avoid? Some probably wouldn't like codes such as 4666, others 1313. What about a pilot not liking 2403 because it happens to be the date a relative of his died? 4 is an unlucky number in Chinese culture, should we avoid all squawk codes with 4 in them?
We use numbers in aviation for all sorts of things, all the time, and it simple does not make sense to avoid some of them for non-operational reasons.