How can I prevent Health Check from deleting my History from CCleaner browser?
You can't.
The settings for what Health Check cleans are fixed and you can't alter them.
Instead you have to use Custom Clean and tell it not to clear the history (and possibly the sessions) for CCleaner browser.
I don't have CCleaner Browser on this machine but you can see here that I have unticked 'Internet History' for Firefox, so Custom Clean will leave Firefox history alone.
Again though those settings only apply to Custom Clean - Health Check always does it's own thing and you can't change that.
You can set Custom Clean to be the default that CCleaner opens to by going to Options>Settings and changing the 'CCleaner Home Screen'.
EDIT- You should also check that the CCleaner Browser is not cleaning the history itself when you close it.
https://www.ccleaner.com/docs/ccleaner-browser/using-ccleaner-browser/cleaning-ccleaner-browser