There really is no reason you can't have both the Count.It app tracking Apple Health and connect another device or app to both track activity for you. Count.It simply takes the best score for your activity on any given day from whichever tracker records it, and we ignore the other one(s). Many people connect multiple devices as backup. However:
If you wish to remove the app in full: As long as you have the Count.It iOS app installed on your phone and you are logged in, it will continue to connect to the Count.It servers and report your Apple Health data (if linked originally). As long as the app is on your phone, it will continue to reconnect to the site - you have to remove the app in full to prevent that.
If you wish to disconnect Apple Health from your account: It is not possible to disconnect Apple Health on your Trackers page (on web or app) like it is with all other apps and devices. You will need to delete the Count.It app from your phone and reinstall. When prompted to connect Apple Health, simply decline and sign back into your account using the same email and password as you used when you first created you Count.It profile. Note: You will not lose any data already reported to your Count.It logs and can get right back in the game without a hitch!