Menu links not working when logged out of Admin but works while I'm logged in

Hi, please see this dev site link : http://www.mediationconnection.net

When I click on "Events," let's say I am taken correctly to the Events page (while logged in to Admin). But when my client tries it he gets a redirect to the homepage. So I tried it in an incognito window in Chrome (Mac) and noticed the same problem - even though "view page source" shows the A link - clicking on the menu item does not jump to that page. I noticed that Mac Firefox does the same thing - broken menu links.

Notes :

1) this is true for all the menu links
2) I have disabled Cloudflare and WPSuperCache at this point
3) I have flushed CF and WPSC as well so that going forward no old files are cached
4) The links work fine for me when logged in to the Admin
5) The links are broken when I'm logged out

Please advise on a strategy.

Thanks so much,

Ed