>>>It was within literally 10 minutes when the customer sent me email saying that they check everything, as far as firewall or AV and were still getting error. And then I started the SQL Server Browser service and asked him to try again. And he said "everything works now." It is, of course, possible that something else may have occurred at the same time. But I don't know.
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>>When the server was restarted, was it still running? Or, when that PC was restarted, was is still running?
>
>It was the Service (not server) that restarted. And it was not running. So I simply went to SQL Configuration and Started this Browser service. PC was running and was not shutdown at any time.

In case it helps you or anyone else in the future, here are my notes on enabling remote connections:

How To Enable Remote Connections SQL Server 2012 Express

Open the SQL Server Configuration Manager
Expand SQL Server Network Configuration – Protocols for SQLEXPRESS
Enable TCP/IP
Right click TCP/IP and open Properties
Check the IP2 settings reflect the servers IP address and subnet
Check IPAll and ensure that Dynamic Ports is blank and make sure that the TCP port is set to 1433
Try connecting to the server from a remote machine using SQL Management Studio or your favourite tool, you should be good!

If not, expand SQL Server Services from the SQL Server Configuration Manager and change the SQL Browser Service to Automatic and start it, then try again.

Also, it’s worth noting that if you’ve already had a fiddle you may need to double check that you have still got Allow remote connections to this server enabled on the instance options. Here’s how to check:

Open Microsoft SQL Server Management Studio and connect to the desired instance.
Right click on the instance node and click Properties
Click on Connections and check that Allow remote connections to this server is checked and you haven’t limited your maximum number of connections over drastically
Warning: When stopping and starting SQL services it is worth making sure that the databases are not in use so as not to effect any live systems.

To make it work, SQL Browser on the remote server may need to be enabled, UDP port 1434 should be put into exception (on the machine running SQL Server) if firewall present.
Also add firewall rule on TCP 1433