redirect Server side with no header User-Agent

You might try "" as one of the switch conditions, but I haven't tested that. This should work though:

when HTTP_REQUEST {
    if ( not ( [HTTP::header exists User-Agent] ) } {
        HTTP::redirect "http://mywebsite/[HTTP::uri]"
    } else {
        switch -glob [string tolower [HTTP::header User-Agent]] {
            "*blackberry*" -
            "*ipad*" -
            "*iphone*" -
            "*ipod*" {
                return
            }
            default {
                HTTP::redirect "http://mywebsite/[HTTP::uri]"
            }
        }
    }
}