A word for someone who intentionally makes vague negative comments and then refuses to explain them

A tease.

1. a person who likes to play tricks and jokes on other people, especially by telling them something that is not true or by not telling them something that they want to know _{(Oxford ALD)}

Beware of sense 3, though.

Passive-aggressive? fr. MW - being, marked by, or displaying behavior characterized by the expression of negative feelings, resentment, and aggression in an unassertive passive way (as through procrastination and stubbornness)

One more try then, since you elaborated further on the description.

Cryptic: having or seeming to have a hidden or ambiguous meaning, marked by an often perplexing brevity

It doesn't really have the negative connotation, maybe a slight negative shading, but otherwise it seems right.

I'd say they were being enigmatic.

enigmatic

adjective

2. having an often intentionally veiled or uncertain meaning. the Mona Lisa's enigmatic smile

There's a book by Suzette Haden Elgin called the Gentle Art of Verbal Self-Defense.

It's not a single word, but she calls this person who makes vague, negative comments a verbal abuser. Her books describe how a verbal abuser may speak injuriously by using vocal inflection.

For example, imagine Tony is having difficulty handling a task. If the abuser said,

Even BILL could handle THAT TASK.

then a native English speaker would immediately know:

1. The abuser does not think much of Tony.
2. The abuser does not think much of Bill's abilities.
3. The abuser does not think the task is very difficult.

Here, the abuser is making a vague, negative statement with little risk of being revealed, because the literal meaning of the sentence is not injurious.