I have done quite a bit of research into this and I simply cannot find much at all about arguments against existentialism aside from determinism, religious arguments and essentialism, let alone any philosophers who were anti-existentialist (I've heard that Blanshard was, but can't find anything about that). All help much appreciated!
One simple example of a philosopher who would have disagreed with the "existence precedes essence bit" is Plato:
Plato's theory of Forms or theory of Ideas asserts that non-material abstract (but substantial) forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality.
He was obviously not contemporary to the modern Existentialists (so not "Anti-Existentialist" in that sense), but his works offer one possible opposite arguments from the Existentialists.
Although he is usually considered an existentialist, (the later) Heidegger is against Sartre's claim that existence precedes essence in his Letter on Humanism:
Sartre expresses the basic tent of existentialism in this way: Existence precedes essence. In this statement he is taking existentia and essentia according to their metaphysical meaning, which from Plato’s time on has said that essentia precedes existentia. Sartre reverses this statement. But the reversal of a metaphysical statement remains a metaphysical statement. With it he stays with metaphysics in oblivion of the truth of Being (p. 250)
'Existentia' and 'essentia' are said of beings as they are ontotheologically thought by metaphysics, i.e. according to the paradigm of an artifact present for the eye. Sartre does not question this paradigm, but merely reverses it, claims Heidegger.
In his earlier Being and Time, which inspired Sartre's existentialism, Heidegger says the essence (Wesen) of Dasein is existence, but he means something different with 'Wesen' (verbally, i.e. the way a being is towards Dasein) and 'Existenz' (restricted to Dasein only: the way in which Dasein is towards itself).