It has negative connotations because it describes the act that is against the law. That's literally what "illegal" means. If you think it's OK to violate the law, then the connotations would be positive, if you do not, then the negative connotations are appropriate. You can't have it both ways.
It's not due to the definition, it's because of how the word has been used. There are terms which refer to undocumented immigrants which still represent their status, but are not dehumanizing.
If a person breaks the law, it is completely correct to say they broke the law. Going out of one's way to hide that fact because stating the facts is somehow "dehumanizing" is a political propaganda exercise, which tries to present (some) laws as something insignificant and barely worth mentioning, and people that consider following these laws important as somehow irrational, and more than once in this very topic - irrational, hateful and racist. I can see why you want to go through this political exercise, but I have absolutely no reason to follow you.
In the United States, literally everyone breaks the law. Calling someone “illegal” as if their identity is defined by something done by literally every single member of society is just a thin veil over the wish to dehumanize the vast and terrifying Other.
By your logic, you are also “an illegal”, because you, too, have broken the law. Everyone is.
But you don’t use it this way. You use it exclusively to define foreigners as bad, as if the law’s opinion about a person’s actions is relevant to their identity.
These supposed "undocumented immigrants" are not generally undocumented. Nearly all of them have documents. Of course, those documents might use your social security number.
This being the case, the situation is generally one of fraud. Some states consider all cases of identity theft to be felonies, while other states can charge the crime either way.
