If you don't want to use extra software, one funny workaround is to connect a Mighty Mouse (not a Magic Mouse) and go to the mouse preferences. The Mighty Mouse has buttons 3 and 4, the scroll wheel click and side buttons squeeze respectively, and you can map those to custom actions. Then you can connect your third-party mouse, and those buttons on it will still perform the actions you set. Haven't done this in a while, so can't confirm it works in the latest macOS. There must be a
defaults write command to do this instead, but I can't figure it out; the mouse prefs files just have integer values for buttons 3, 4, and 5 that don't seem to affect anything when changed.
If you just want the extra buttons to control Exposé/Mission Control, you can do that in the Mission Control prefs without connecting a Mighty Mouse.