It is a complicated legal question.
To your specific point my counter argument would be that the section you quote elucidates specific rights that shall not be abridged but does not extend to the broad scope of rights that most citizens enjoy.
I say "most" because there are exceptions even for citizens such as the limitations on minors, those on felons, those with mental and/or physical disabilities etc.
In this case I would argue that illegal aliens, by virtue of their hostile presence in our jurisdiction, enjoy those basic rights enumerated above in the 14th A. but very little more. They are handicapped by a complete absence of legal standing to be in U.S. jurisdiction at all.
I know there is legal precedent for legal aliens to purchase and possess firearms but I am not aware that such exists in regard to illegal aliens. There may be.
They are entitled to the equal protection of the law but the question is does the law allow the possession of a firearm by someone who violated the sanctity of our jurisdiction in the first place? Someone whose very presence is a direct violation of the boundaries of said jurisdiction?
It's sort of a "Starship Troops" kind of a thing. Citizen vs. civilian.