Secret Service kinda gets a little hooked up, because the agency still has a travel budget they can draw from if they travel, which is common for them. Their regular paychecks though? Nope.
Ceterum censeo Carthaginem esse delendam
You're right, with the VA and Defense Department still receiving pay I thought certain federal employees were deemed super necessary and thus still received a paycheck during a shutdown, but apparently it's just because they are funded differently but if the shutdown lasts long enough (which is unlikely) they will stop receiving pay as well.