I've never actually used the champion missile module.
They have excellent range, very good firepower, good fire rate, and you always start with them. All in all, IMO, one of the top Champion modules.
You lose a shield slot for them though, and the starting human champ has not a ton of those. I tend to just stick with shields, and then by the time i could do pretty good with missiles I could do well with several other things and they're sorta forgotten about.
The base Frigate health is 15,000,000 HP. With 2 Mk II Shields, that's another 15,000,000 health for a 100% increase.
Base Frigate damage is around 250,000 DPS (assuming Mk II modules and not applying multipliers).
Switching over to Mk II Missiles gives an increase of 120,000 DPS against everything (+50%) and a huge 600,000 DPS increase (almost 250%) against Starbases.
Because Starbases are the primary target in 6 of the 9 scenarios, Missiles seem a good choice, over more shields.
As an aside, when I was doing some Champion vs H/K testing, I found that designs with more shields actually performed worse than higher damaging designs. I think it is because the Shield modules for Champions are so weak. As such, even a Spire BB with all shields only gains about 170,000,000 HP, or three times its base HP. This probably is the reverse result when using small modules, like Flak, against fleetship hordes, though.