I think you may need to do is to serve your CDN assets over HTTPS. To do this you'll need to enable HTTPS within the Azure Portal
if still not then:
You can't yet use your own domain/SSL certificate with Azure CDN although there is a big request for this on the Azure Feedback Site.
If you've created a CDN for an Azure Cloud Service (e.g. http://[yourapp].cloudapp.net/cdn/) it's important that you create a self-signed certificate for your Azure domain ([yourapp].cloudapp.net). If you're using Azure Virtual Machines this can be done through IIS.
This caught us out recently when we enabled HTTPS for an origin server that did have an SSL certificate but it was for our own custom domain and not the *.cloudapp.net one.
for details click on http://benfoster.io/blog/azure-cdn-https