Just in case anyone else would encounter the same roadblock in the future, I solved this by accessing the instance variables directly like so:
First, require the httpbinding client, instead of client
require 'xmpp4r/httpbinding/client'
then connect by doing
@client = Jabber::HTTPBinding::Client.new("your_jabber_id_with/resource")
@client.connect("your_bosh_url")
@client.auth("your_jabber_password")
The instance variables are @http_sid and @http_rid, and since they are not accessible I did
sid = @client.instance_variable_get("@http_sid")
rid = @client.instance_variable_get("@http_rid")
Then I stored those variables in my session so that I can use them on my Strophe.js client.