This can be done with the HTTP Request and the Internet Request functions in the Flic Apps.
But basic auth does not seem to be supported. So in order for me to call my Fibaro rest API, I have had to setup a node proxy between my flic buttons and the Fibaro API (which requires auth)..... :/
Completed since long!
I agree, this is what I need to control my Fibaro home automation system with a bunch of flic buttons around the house...
I second that.