@Peter - Thanks, but I think you missed the question, URL encoding of = is not the issue, the issue is using single quotes in the where statement.
The statement I'm trying to run (which does not work) is: where=ShortTitle = 'My name''s Tom'
The statement that does work is: where=ShortTitle = 'My Name Top'
The addition of the two single-quotes makes the service return a 403 error.
BTW, the URL was being encoded properly but just ended up in the link I copied from the Chrome's network tab, the only thing that didn't get converted when I pasted was the = sign. Using %3D didn't change anything.
Encoded versions of the where statements:
where=ShortTitle+%3D+%27My+name%27%27s+Tom%27 (doesn't work, returns 403)