Setting up SSL - Will it solve my problem?

My website is used by my customers to provide a live chat facility on their website. They do this by pasting a javascript tag from my system on their website, which activates the paid-for features on their site. One of my customers uses SSL across the entire site and as such gets warnings on every page load in IE stating that one or more of the items on his page are on a non-secure server (i.e. my script). I assume I need to get SSL set up and a certificate, but being that I don't know much about SSL, I'm concerned that his customers will simply get a different popup, this time asking if they want to accept my SSL certificate, which obviously would defeat the purpose of getting SSL setup when all I'm trying to do is stop his website from bugging his customers with an SSL-related popup.

What do I need to know in this regard before I go and order the SSL add-on and buy a certificate to go with it? I have no idea about this, so any advice is appreciated.