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You stopped reading two seconds too soon: 

You are welcome to use FileRun for both personal and commercial purposes, at no cost.

For personal/family use (up to 10 users), or for registered non-profit organizations,
contact us with your FileRun installation URL and we'll do our best to accommodate your usage, at no cost.

Then it's not about the domain name as the topic title says, it's about the protocol. You should look into configuring your reverse proxy server to forward the right HTTP headers, to preserve the external protocol used.

You just have to delete the file "system/data/filerun.dat", access FileRun via the new domain and re-register from its control panel.

Using the forums automatic post searching will get you answered to common questions like this one. ;)