Terminology - What is the exact word to describe 'being skeptical of something/someone'?

I think you are looking for paranoid.

Paranoia

baseless or excessive suspicion of the motives of others.

Suspicion and mistrust of people or their actions without evidence or justification.

In modern parlance, you'd say Jack was a blamer. Usually found alongside guilt and shame/embarrassment, so a blamer spends a lot of time avoiding feeling guilt and shame by making everything someone else's fault, or anyone's responsibility other than his own. Persecution complex might fit, but, together with paranoia, are actually mental health problems. At the very least, poor old Jack is inadequate.

You could consider the term scapegoater.

While this exact word does not appear to be listed in the stock online dicitonaries, it does appear in literature, beginning around 1910, and increasing in usage after 1940, as evidenced by this ngram.

The meaning of the much more used term scapegoat is

a person who is blamed for the wrongdoings, mistakes, or faults of others, especially for reasons of expediency.

The verb form is defined as

make a scapegoat of: secret police scapegoated a few of the ringleaders to put an end to the issue

Jack's actions could be characterized as scapegoating

They also could be considered projection, a psychological concept defined as

the attribution of one's own ideas, feelings, or attitudes to other people or to objects; especially : the externalization of blame, guilt, or responsibility as a defense against anxiety

It also sounds like Jack considers Jill his nemesis

a long-standing rival; an arch-enemy:

Answering the question in the title:

cagey (cagy)

1) Wary; careful.

If I say something about it, I'm ostracized. I'm a leper. So, that's why I'm cagey about this with you. / a cagey avoidance of a definite answer

leery

Slang. (followed by of)

1) Suspicious or distrustful; wary. Openly distrustful and unwilling to confide.

was leery of aggressive salespeople / I'm leery of his financial advice. / They are leery of the proposed system. / I too was always leery about work at home offers because they always seemed to be scams.