The Proofpoint Essentials service will continue to try and check the customer's mail server, as per the standard set-up. The Emergency Inbox facility runs alongside the normal mail flow and mail delivery/error handing.
If Proofpoint is unable to relay the email from their system to the customer's server (as it is down), they are required by RFC to send a deferral notice back to the sending mail server that they've not been successful in delivering the email.
As long as the customer's mail server is down, they will continue to send deferral messages for some time (set at 14 days) then bounce the message back to the sender. It is important that the customer mail server be brought back as soon as possible to not lose email.
More detail including Emergency Inbox access help here.