Actually, Everything Search can search on the network, but you have to create a file list yourself.
Go to Tools -> File List Editor
Click Edit -> Add folder --> here, select a shared folder on the network and hit OK.
Everything Search might freeze and it might take from 1 to 10 minutes or longer to create a list so be patient.
When you see files in the list editor, click File -> Save As... - and save the file somewhere.
Close the File List Editor, in Everything Search main window, click Tools -> Options, from the left hand menu select Indexes --> File Lists, click Add button, select the file that you saved previously, Apply settings and that's it.
From now on when you search it will display results from the network as well.
You can automate the file list creation and schedule it with Windows Task Scheduler. Just create a BAT file with the following:
"C:\Program Files\Everything\Everything.exe" -create-filelist C:\file_lists\filelist.efu S:\
Modify the code above to save the file list where you want (C:\file_lists\filelist.efu) and specify your own drive to index (S:/).
Then in Task Scheduler create new task to open this BAT file for example once a day.
Then make sure you point Everything search to the file you create with BAT file.
(Tools -> Options, from the left hand menu select Indexes --> File Lists, click Add button, select the file (C:\file_lists\filelist.efu), Apply settings)