On the privacy of perfect hash functions
The definition of perfect hash functions do not have any security/cryptographic requirements. For example, the hash function that simply outputs the first n bits of any string is a perfect hash function on the set of n bit strings. Obviously this function does not hide the input at all. 

So the answer to your questions is that it depends on the hash function. It being a perfect hash function alone (minimal or otherwise) does not guarantee any privacy at all. Neither does it say anything about what happens outside of the set for which the function is perfect. That means if the users hashes a value outside of this set it may well result in a collision with value in the set. Such a collision is easy to construct for the hash function I mention above.

On the other hand, if the reason you want to use a perfect hash function is that there are no collisions, you might as well use a cryptographic, collision resistant hash function instead. In this case there is basically no way for user to find a value outside of the set that collides with one in the set. 

If using a collision resistant hash function in your protocol is enough for secure private set-intersection is a more complicated question. It essentially depends on your security assumptions and the flavor of PSI you want: Is the server malicious? Is the client? Should both parties learn if there is an intersection or not?