I have my ceiling fan disassembled to the point where I have separated the light section off the main fan. The broken chain is inside a half-black / half-clear case with a couple of tabs holding the two halves together. Is there a "trick" to opening that case? I'm concerned that if I'm too hard the tabs will break off and it won't stay together (unless taping it back together could be acceptable)

No trick but easier to replace the switch. Fan switches are pretty generic, make sure to get a 3 speed fan switch.

So crack the case open to get to the four wires and the chain? The new switch will have the chain with it? Thanks!

The new switch will have a chain and 4 wires. Cut the old wires and wire nut together.