<p>I have an <a href="https://en.wikipedia.org/wiki/Microsoft_Office#Windows_versions" rel="nofollow noreferrer">Office 2007</a> (specifically <a href="https://en.wikipedia.org/wiki/Microsoft_Outlook#Outlook_2007" rel="nofollow noreferrer">Outlook 2007</a>) add in created in Visual&nbsp;Studio&nbsp;2008.</p>

<p>When I uncheck the "Sign the ClickOnce manifests" option, and then publish, it rechecks that option automatically.</p>

<p>I have a regular <a href="http://en.wikipedia.org/wiki/Windows_Forms" rel="nofollow noreferrer">Windows Forms</a> project in the same solution that allows me to publish with this unchecked.</p>

<p>Why isn't it allowing me to uncheck the sign option?</p>

You are not required to sign EXE files, but you need to sign DLL files for deployment. Your Office add-in is in the form of a DLL file, so it must be signed. Your Windows Forms project is linked into an EXE file, hence it does not need to be signed.
See the relevant MSDN page,
ClickOnce Manifest Signing and Strong-Name Assembly Signing Using Visual Studio Project Designer's Signing Page
, sub section
Signing Assemblies
.