What do you mean by
shiny app examples where they were launched on HPC
? If you want the R process supporting the Shiny application to run on HPC, then this is currently not possible. That's what I meant with "no direct connection". This is different to RStudio Server Pro, where it is possible to execute the R session itself on a cluster (with Slurm and Kubernetes being provided out of the box).
However, you can have the R process supporting the Shiny application run alongside RStudio Connect and within that R process start an HPC job either "manually" (write batch file and call system()) or via a package such as batchtools. Have you seen the webinar Scaling Data Science with R at Janssen Pharmaceuticals - RStudio? Between minute 18 and 21 such an example is presented.