Am curious about this method which I have never used...
Following your recommended steps, Q about these steps...
3, 4. Assuming that the first machine is the machine without the desired Hosts file and the second machine is the one with the desired Hosts file, is this really any different than simply booting the first machine naturally but with the second machine's disk mounted as a second drive?
5. Not clear what is meant by "access," are you suggesting more than or only reading the second disk's file system?
Unless I'm missing something,
1. mounting the FreeBSD disk as a second drive won't work unless <maybe> something like FUSE might support if there isn't native support for the secondary disk's filesystem. This means that after investigating the FUSE option, network file sharing seems to be the most likely option. BTW - it might be worthwhile to consider filesystem support in <both> directions. So, in this case if you can mount the FreeBSD disk while itself booting or as a secondary disk in another FreeBSD instance, consider also mounting your Linux disk as another secondary disk...That should grant you read access to all your mounted disks, ie Although Linux might not support the FreeBSD fielsystem, perhaps FreeBSD will support your Linux filesystem.
2. Once the file system is accessible, seems to me it should be trivial to just copy and modify the template Hosts file.