Nope, you don't need to redirect anything.
It could be that you are using single quotes instead of backticks. Since you're using bash, you should use $(cat servers) instead.
If you're looking for other tips. The line beginning with SERVERS= can be removed as you reset the variable in the following line, and I would not use a pluralized variable for a single item. One last thing, printing out the name of the server you are connecting to can make things much clearer as well. Here's a quick rewrite.
Code:
#!/bin/bash
for SERVER in $(cat servers)
do
echo $SERVER
ssh MyUserName@$SERVER df -h
done