Hi,
The model will need to have a Raster Statistics node between the file input and the Root Stretch Raster node. The latter node can't connect directly to the file input, nor can it determine min/max values automatically from the file input. It will need the Raster Statistics node in order to extract the min/max values of each band.

However, you shouldn't have to manually enter vectors of mins/maxes. When you connect the Raster Statistics node to the Root Stretch Raster node, map the "Minimum" parameter of the former to the "Minimum" parameter of the latter. Same with the "Maximum" parameter. I wish I could show you a screen capture, but we can't attach images here.

As far as the data ignore value, I just created a simple model similar to what you are creating, and tested it in ENVI 5.6. My sample image had the data ignore value set to 0. When I ran the model to produce the square root stretch raster, the result correctly honored the data ignore value. When I right-clicked on the result in the Data Manager and selected "Quick Stats," it gave the correct data minimums while ignoring values of 0. Can you provide a few more details on what you saw that was different?