The on-loan Manchester United shot-stopper was slapped with the suspension after being found guilty of improper conduct in the 3-1 defeat at Blackburn Rovers earlier this month.
Objects were thrown towards the goalkeeper and other Town players, and it is suggested Henderson threw them back into the crowd.
Town finally received written reasons on Monday from the FA, and chief executive Brian Caldwell and boss Paul Hurst decided that an appeal was not worthwhile, noting the risk of adding extra length to the suspension.
Henderson has already served one match of the ban, as Craig MacGillivray deputised superbly in the 1-0 win at Portsmouth.
Caldwell told shrewsburytown.com: “I spoke with Paul about appealing and we don’t think it’s worthwhile. You look at the process and think, there’s a potential it could add another game or games if we did appeal.
“We have to abide by the rules and make things better going forward somehow.
“We got the written reasons yesterday. They’ve decided, the way I understand, on the basis of probability. If there’s a probability that something happened and they think it has happened then the commission then have the right to decide yes or no.
“It’s not like going into a court where you’re innocent until proven guilty. There are certain things with the process I’m not happy with and I flagged that up at the weekend, I sent an email looking for answers in how evidence is gathered.
“We’ve also been keen to see if Blackburn have been charged in all of this. While you can’t condone anything that happened, two of our players were pelted (with objects).
“I’m sure the FA have looked at that and would be keen to improve how it has been handled going forward.”
Henderson will also miss Saturday’s trip to Bristol Rovers and the following weekend’s visit of Plymouth.
Caldwell and the club were unhappy at the handling of the situation. Where Henderson’s ban was registered and publicised before the club had received written reasons and were unable to make a stance.
He added: “It’s happened twice with the FA Youth Cup too. You get the decision with what’s happened but not the written reasons.
“The FA aren’t obliged to give the reasons for 48 hours. We’re at a situation where we’re releasing statements about his punishment and then I understand everyone is wanting to know what we are going to do.”