Filestream database backup error
FindNextFile is a Windows API call, so what's happening is the backup process is enumerating all the files in the FILESTREAM folder and backing them up. The error message indicates that the operating system cannot read the file system; this indicates, at least to me, that the file/folder structure on disk is corrupt. There would be a different error returned if the file/folder was simply not found. This may mean the I/O subsystem has problems. I would run chkdsk in Windows on that volume.

From the DBCC CHECKDB documentation, it doesn't look like FILESTREAM checks go beyond ensuring there is linkage between the database and the file system. Running chkdsk has the ability to do a low-level check of the disk contents itself, which is probably what's needed in this scenario. It certainly wouldn't hurt to run CHECKDB on the database anyway (read: regularly) either.