Is this the same computer where the files started out in CS5?
There are probably quite a few different cuts of Avenir (certainly over the years it has been available as TT and T1 in both Windows and Mac formats, and I have an OpenType version here from Linotype). Some possibilities for why CS6 might not see a font are that you have installed a different version, you have more than one version installed or it is installed in more than one location, or that it is looking at a different internal name in CS6. Typically, if the fonts are showing as missing, but are, in fact installed, using Find Font and substituting the same font will solve the problem.
No, our advertising company created the files and saved them to an external which I have. The problem arose when I sent the files to our printer to have them printed and it was missing all of the links. But when I open it on my computer, only some of the links are missing. On my end at least since I've been looking at it for awhile-I think Avenir Black is showing up as Avenir in InDesign. Something I don't understand is that after I replace Avenir Book with itself, save and re-open I'm having the same problem.
This is not something I think I'm going to be able to figure out without seeing the file and the computer where the fonts are installed, which is not possible unless you happen to be located someplace on Cape Cod. Sorry.