There are clearly carved-out exceptions to that, Salvation. Your parish is free to hire Catholics only. A Presbyterian church is free to hire only Presbyterians. Any other outcome would be absurd.
Wheaton does not specify denominations. It can't--it's an endless task because their constituency is so varied and some of Wheaton's constituents (Plymouth Brethren) would be offended by specifying denominations at all--it would exclude them. So Wheaton chose the route of using a doctrinal statement instead of church membership as the criterion. But Evangelicals disagree about so many aspects of doctrine that the statement had to be left vague.
Now, in practice, they have added as a gloss to their doctrinal statement a negative-denominational criterion: one may not be a Catholic even if one can sign the doctrinal statement in good conscience. Most of their constituency will think this negative-denominational criterion reasonable and that's why they added it. But I doubt that it was spelled out contractually--perhaps there was a clause like that in contract boilerplate.