First, I am not an expert on this. Let me give you the names of three experts: Noam Elkies, Abhinav Kumar, Matthias Schuett.
The first result that you want is so strong that it must be open. I am pretty sure that it is open even in dimension 2 -- K3 surfaces -- so you should probably start with that. (I don't see how a reference could establish that a problem is open. At best, it was open as of such and such a time. Better to ask the people above.)
Maybe the right question to ask is what is the most general class of K3 surfaces which can be shown to have infinitely many primes of supersingular (resp. ordinary) reduction? I think the case of singular K3's (i.e., with Picard number 20) should be the easiest due to connections with the theory of complex multiplication. I expect that the result is probably known in this case, at least for certain CM types. You should also look at Kummer surfaces because of the connection to abelian surfaces (is it true that supersingularity/ordinary passes from the abelian surface to its Kummer surface? it seems plausible). There are a lot of results on primes of ordinary reduction for abelian varieties: a conjecture of Serre is that, after a finite base change, the density of primes of ordinary reduction is always equal to one, and a lot of special cases of that conjecture are now known (e.g., possibly for all abelian surfaces).
Some of the above should generalize to Calabi-Yau's with complex multiplication, I think.
It's a very interesting question: please let us know what you find out.
ADDENDUM: Here's an off the cuff idea to show that the problem must be open: start with an elliptic curve E over an imaginary quadratic field K. Let A be the Weil restriction from K down to Q, an abelian surface. Let X be the Kummer surface. As above, I am guessing that X is ordinary/supersingular iff E/K is, and this is well known to be an open problem in general: there are some examples due to Elkies and Jao where infinitude of supersingular primes can be proven, but very few.