Like the description says: The "Upgrade Equipment" button is greyed out when you equip your soldiers with a non-upgradable weapon like the Darklance or a vanilla autopistol (Templars)

This seems to be a known bug for quite some time. Just wanted to put this here so noone has to search as long as I did.

Possible solution for accessing the upgrade slots nonetheless is Robojumper's Squad Select or switching to another soldier, open "Upgrade Equipment" and then switch back to the soldier in question.