Windows Shared Storage / External SAS3 / JBOD

Then when I went to the other server, it showed the drives as "Foreign".

This is because NTFS (and ReFS) are not cluster file systems. The is no way to access a volume from two windows machines at the same time. What seems to be very easy when using Objectstorage oder vmWare VMFS actually needs a lot of magic under the hood.

Is it possible to create a mirror set and have both servers access it?

No. Even in a Windows Server Cluster scenario, the drive would failover from one host to the other, instead of beeing used by both machines at the same time.

I know I could put the drives in a NAS enclosure and accomplish it.

In that case, the drives wouldn't be accessed by a file system driver (like NTFS or ReFS), but by a network protocol (like SMB oder NFS), which is designed to allow multiple access.