2 Further points to make
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Edit: JavaLatte points out that your question is actually about the adjectives un/disqualified, not about the verbs. This is important because (as my answer explains) the use and meaning of the prefixes is very different between verbs and adjectives.

The prefix un- with a verb generally has a meaning of reversing or taking back some action. Because most actions in the real world can't actually be reversed once they have been done, before the advent of computer interfaces the prefix was more or less restricted to verbs of wrapping, enclosing, attaching: unlock, unfasten, unwrap, uncover. There were exceptions, but they were very rare.

In the virtual world of computer interfaces, there are many more actions which can be reversed, and so the prefix un- has recently become much more productive: words like undelete, and unsubscribe have become common.

I have never heard unqualify, but I would take it to mean "remove the state of being qualified" (or the corresponding unaccusative sense "cease to be qualified"). "Disqualify", on the other hand, could have this meaning, but could also mean "Prevent from ever being qualified".

In the particular instance you ask (an unqualified product vs a disqualified product) I would understand that an unqualified product has never been through whatever process is used to qualify products: it may qualify, or it may not, but it hasn't been evaluated. A disqualified product is one that has failed to qualify.

The prefix un- with a verb generally has a meaning of reversing or taking back some action. Because most actions in the real world can't actually be reversed once they have been done, before the advent of computer interfaces the prefix was more or less restricted to verbs of wrapping, enclosing, attaching: unlock, unfasten, unwrap, uncover. There were exceptions, but they were very rare.

In the virtual world of computer interfaces, there are many more actions which can be reversed, and so the prefix un- has recently become much more productive: words like undelete, and unsubscribe have become common.

I have never heard unqualify, but I would take it to mean "remove the state of being qualified" (or the corresponding unaccusative sense "cease to be qualified"). "Disqualify", on the other hand, could have this meaning, but could also mean "Prevent from ever being qualified".

Edit: JavaLatte points out that your question is actually about the adjectives un/disqualified, not about the verbs. This is important because (as my answer explains) the use and meaning of the prefixes is very different between verbs and adjectives.

The prefix un- with a verb generally has a meaning of reversing or taking back some action. Because most actions in the real world can't actually be reversed once they have been done, before the advent of computer interfaces the prefix was more or less restricted to verbs of wrapping, enclosing, attaching: unlock, unfasten, unwrap, uncover. There were exceptions, but they were very rare.

In the virtual world of computer interfaces, there are many more actions which can be reversed, and so the prefix un- has recently become much more productive: words like undelete, and unsubscribe have become common.

I have never heard unqualify, but I would take it to mean "remove the state of being qualified" (or the corresponding unaccusative sense "cease to be qualified"). "Disqualify", on the other hand, could have this meaning, but could also mean "Prevent from ever being qualified".

In the particular instance you ask (an unqualified product vs a disqualified product) I would understand that an unqualified product has never been through whatever process is used to qualify products: it may qualify, or it may not, but it hasn't been evaluated. A disqualified product is one that has failed to qualify.

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The prefix un- with a verb generally has a meaning of reversing or taking back some action. Because most actions in the real world can't actually be reversed once they have been done, before the advent of computer interfaces the prefix was more or less restricted to verbs of wrapping, enclosing, attaching: unlock, unfasten, unwrap, uncover. There were exceptions, but they were very rare.

In the virtual world of computer interfaces, there are many more actions which can be reversed, and so the prefix un- has recently become much more productive: words like undelete, and unsubscribe have become common.

I have never heard unqualify, but I would take it to mean "remove the state of being qualified" (or the corresponding unaccusative sense "cease to be qualified"). "Disqualify", on the other hand, could have this meaning, but could also mean "Prevent from ever being qualified".