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When I login to iTunes or Skype the fonts all show up as boxes.
No fonts were removed on installed; none are disabled and all the normal ones show up. Doesn't happen on any web site; only seems to be locally installed programs.
Uninstalled and reinstalled and programs still don't work.
This usually happens when you have Fonts Caches (or actual Fonts) that are corrupt. I would do the following, to not only clear the caches, but also to verify your Fonts.
Clearing the font cache - you have a few options (Onyx is just easier).
atsutil utility. You basically would do astutil databases -removeUser then sudo astutil databases -remove, then astutil server -shutdown, then astutil server -ping. I would recommend restarting after this./Library/Caches folder, but I am not sure how well this works. There also used to be some older ways to clear the cache but I cant find them (and I am pretty sure it was tools before Mac OS 10.5). Verifying Fonts
/Applications/)Validate Fonts. 
Usually clearing the Font Cache works. I used to see a similar problem all the time with Microsoft Office 2004, and it had its own cache that we even cleared weekly. I haven't seen the Mac OS mess its cache up that often, but when it does, usually a quick fix. I have even seen a reboot sometimes clear it up (in very minor cases).
I think jmlumpkin nailed it with clearing the font cache.
For what it’s worth, there’s a much easier way to clear the font cache for many people that doesn’t involve the Terminal, and that is by using Safe Boot. Once booted to the login screen, you don’t even need to sign in — simply restart, and use the computer normally.
