If guest click on user antony he get redirected to userprofile/antony
That's perfectly normal. Clicking a users formatname outside of profile always links to their profile.
do we need to set menu to public access
If you don't want public profile access then turn off public profile access in CB > Configuration > User Profile by adjusting the profile access parameter.
is it secure?
Yes. I don't see what issue you're having or what this has to do with redirect on login.