I've seen multiple mentions on this site about how bad "backstab" connections are. I recently replaced a switch for a fan with a timer, and the switch had backstab connections instead of screws (the timer has screws). That switch looked identical to all the other fan and light switches in my house, so I'm guessing they have the same type of wiring connection. Should I be worried about this? Do I need to replace them or do anything else to prevent fires? Or is this more important for outlets than switches? ... Should I check my outlets in case they're the same?
95% of the time, backstabs fail "open" - they seem pretty good at that. That's not a fire starter, but it is an annoyance that will send you on a day-long "bug hunt" for the faulty connection.
The other 5% of the time, it will create a weak, overheating connection or a series arc fault, and yeah, that'll start a fire.
The main reason we recommend scourging them is the future annoyance of a failed "open".
Should I be worried about this?
Not necessarily. Depends on how much you like to worry about things :-)
Do I need to replace them or do anything else to prevent fires?
If your home is fairly new (or if the main panel was replaced or upgraded recently) then you may have AFCI breakers, which can provide quite a bit of protection against flaky connections, including backstabs. If you don't have AFCI then this is more of a real (but minor) concern.
Or is this more important for outlets than switches?
I would be more concerned about outlets than switches. The stress on a switch in normal use is pretty much "none". The stress on an outlet can be considerable as items are plugged & unplugged. That stress/vibration/movement is what leads to failure of backstabs. In addition, a light or fan has a pretty constant, generally small (except for bathroom heater fans) load. A receptacle might have a tiny load (e.g., a cell phone charger) one day and a large load (e.g., vacuum cleaner, toaster, hair dryer) the next. So you may have a marginal connection that is not a problem at all with a small load but becomes a real problem with a large load.