Feel free to play any game with any switch you like.
HOWEVER... Switches with hysteresis (MX clicky) or hard reset bumps (Topre and MX Clears) make it harder to do certain moves in FPS games, assuming you can actually do them. With linear MX switches you don't have to move your fingers as much to actuate / deactuate switches (by keeping them partially pressed, "hovering"), so you can be a little quicker with changes in movement. MX Browns almost allow this, but I find I tend to go just past the bump / just above the bump. There is a linear section before the bump, so you get close-ish to the actuation point. It's good enough for me (very casual gamer who doesn't like the feeling of typing on linear switches), but not some others. Reds and Blacks really do allow super fast changes as long as you are used to where they actuate.
This is why I say that Browns are a good compromise, but if you are really into gaming AND want a nice board for typing, get 2 keyboards. I find I game okay even on Clears, but if I'm feeling just the slightest bit competetive I'll switch to Browns. Considering putting together a board with Reds just for gaming, though. It really depends how important gaming is to you.