No, you do not need Exchange to use EIOBoard. This applies to both EIOBoard-Hosted and Customer-Hosted versions.
If you still use the Outlook client (the thick client, not the thin/web client) then you will still be able to use the EIOBoard Outlook Interface and perform calendar synchronizations with it or the EIOBoard Application Interface. This is possible because we do not synchronize directly with Exchange, but rather the Outlook mail profile on the local PC.
If you do not use Outlook you can still use the EIOBoard Application or EIOBoard Browser Interfaces, you just will not be able to sync your calendar events.
The same answer applies if Exchange is hosted externally, such as with Microsoft Office 365. It is also true with other cloud services with email hosting such as Google.