This should be straightforward I hope.
The way you would achieve this is to use a 'request rule'. This is a rule that is run by a virtual server each time Stingray receives a request, and you can easily do things like inspect the request and issue a redirect.
You can create rules in two ways - by using the RuleBuilder, and by using the more sophisticated TrafficScript language. Rulebuilder rules are simpler and easier to understand; here's one that should do what you need:
To apply this:
Create an HTTP virtual server listening on port 443, and enable SSL decryption with an appropriate certificate. Load-balance traffic to your target servers (optionally re-encrypting if necessary in the pool)
Verify that you can browse your service through the virtual server.
Create the rule as described above, and apply it as a 'request rule' to the virtual server configuration.
You should observe that any requests for '/' are immediately redirected to '/everywhere'
There's some good gettingstarted resources here (Getting Started with Stingray Traffic Manager), and you can do a quick search for 'redirect' on Splash to find many other more sophisticated examples.
Hope that this helps you get you started - best regards