Chrome (or any other browser) will not allow a HTTPS secure webpage to be displayed in a HTTP webpage.
So, Both need to be HTTPS.
myserver would need a security certificate added to it for HTTPS.
I appreciate the suggestions guys. I went down the SSL rabbit hole for several hours without any luck, but I dug deep enough that I eventually figured it out anyway. It turns out Home Assistant made a breaking change several versions back blocking the use of iframes to prevent clickjacking. This is obviously going to be a problem with the webpage component in myDesigner.
For the sake of posterity if anyone here ever wants to access Home Assistant from within a myServer template you'll have to modify your configuration.yaml file and add the lines:
http:
use_x_frame_options: false
I still have some formatting work to do to pretty things up but I've already tested it as a proof of concept and I think this is going to work out. Thanks again!