Can i buy seperate tickets for each leg or do the stops all need to be on one ticket?

Amaya, it is the itinerary that makes you qualified and not the tickets. In other words, you can use completely separate tickets and travel on unrelated airlines as long as your flight itinerary (route) follows the pattern of A country - B country (mainland China) - C country.

So, your idea or better to say your itinerary:

Budapest - Beijing - Seoul

Is definitely and undeniably eligible for 144 hour visa free transit in wider Beijing area regardless of separate tickets. You should just take care that you have your flight from Beijing to Seoul booked even before your departure from Budapest. Also, I strongly encourage you to use correct terminology because this is not the 144 hour visa, but 144 hour visa free transit, so there is no visa involved at all.