What classes of aircraft require hydraulic flight controls?
Hydraulic flight controls are not mandatory for a specific class, but above some size and speed it will be hard to fulfil all certification requirements without at least adding some hydraulic boost.
Historically, aircraft used manual controls. With increasing size and speed, the forces went up and increasingly clever force reduction systems were invented to keep stick forces within limits.

On the first flight of the Messerschmitt Me-321 glider it was found that control forces especially in roll were way too high, so a second pilot with a second stick was installed.
The Dornier Do-335 or the Republic P-47 used a telescopic stick: At low speed the stick was short and could move through the full range of aileron deflections while at high speed it could be telescopically extended. This limited the motion range but improved the leverage of the pilot.
The English Electric Canberra used internal balances for the ailerons and a spring in the aileron controls to limit the stick forces. At top speed the control horn could still move through its full range but the achievable aileron deflections were only a few degrees - still enough for sufficient roll agility, however.

Ailerons have mass. There is a spring. Something that moves. Every engineer should think next: What about flutter? With more complex force reductions, the chance grew that something unexpected happens and failure modes multiplied. In the end, introducing hydraulics solved so many problems that de facto above a certain speed and size combination no other option makes sense.
I have worked together with Rockwell (formerly North American) on a two seat jet trainer; a size and speed combination that would have had manual controls 60 years ago. I proposed to use the same force reduction mechanisms that were used on the F-86, but in the end a hydraulic boost was selected. Why?

Nobody had any experience with manual controls anymore. After so many years, all F-86-era engineers were long retired or had passed away.
The North American archives had been thrown away years before, so no documentation was left on how to properly design a manual control system for that speed range.
The hydraulic boost system was easy to install, used standard components and was cheaper to design because everybody was familiar with the technology.

As I said before: Don't get me started on how exactly you can tailor roll stick forces - this will be a post that will be much longer than anything I have written here before.