It worked!
First I called tech support about my iPhone getting the error (User is not allowed access to this share) and they said that they were unaware of any such issues. He very politely told me that it must be something Apple did when they updated my phone because he and the engineers have been very frustrated with Apple changing things and constantly breaking Buffalo product features. However, the iPhone has had this problem since I got it 8 months ago!
So a few days after the call with tech support, on a hunch, I came here to this forum and found the solution!
I added the folder 'share' to my BuffaloNAS and gave it Anonymous access rights and bingo, the iPhone is now access files!
:smileywink: