I came to Hungary (from the US) for family business and during my stay here I attained Hungarian citizenship. However, I have overstayed my 90 day visa. When I return home to the states, what will happen to me? Will I be okay if I showed my Hungarian identity card with my US passport when I leave?
First, if you were solely a US citizen before, then you shouldn't have been able to obtain a "90-day visa" for Hungary. That would be a short-stay visa according to the Schengen rules, and such visas are not issued to Americans. Instead US citizens are allowed short stays in the Schengen area without any visa as long as they are not present in the area for more than 90 out of any 180-day period.
However, the 90/180 day rule stops applying to you the moment you acquire Hungarian citizenship. The Hungarian citizenship allows you to be in Hungary (and, within rather liberal bounds, any other EU country) for as long as you desire, no matter what happened to you in your previous life. You cannot "overstay" the 90/180 day rule because it does not apply to union citizens.
I'm guessing that you haven't gone to the trouble of getting a Hungarian passport yet. That shouldn't be a problem, however. In contrast to the US, the Schengen countries do not require dual citizens to use "local" passports when entering or leaving the area. So you're completely allowed to leave the area based on your US passport.
You will want to carry (and present) some sort of proof of your Hungarian citizenship, however, since that its what shows you're not an overstayer. Perhaps a Hungarian identity card will work for that (does it explicitly state your citizenship? If so, you're probably fine); perhaps you need to carry more formal paperwork.
You would probably minimize your risk of delays and other trouble if you can arrange to leave the Schengen area though a Hungarian airport (that is, no layovers in other Schengen countries), such that the border guards who process you will know what the appropriate paperwork looks like and can read Hungarian if necessary.
Your trip will involve a layover in the EU as there are no flights direct to the USA. You probably will transfer in one of Vienna, Rome, Zurich, Munich, Frankfurt, London (yes, you can go through Dubai but don't do that not because of immigration but because it takes way too long). Within the EU your Hungarian ID (which should be a plastic card) is perfectly fine for travel, no need to show the US passport to anyone. When you board a plane to the United States, at check-in show your US passport otherwise they won't let you board. As a US citizen a) you must use your US passport to enter anyways b) you are not eligible for ESTA so your Hungarian credentials are useless and unnecessary to enter the USA. At passport control within the EU show your Hungarian ID, however. If it's such a flight to have a US specific check at the gate before boarding, show US papers. It's always at the gate and as such it's easily recognizable -- you can't mix up a passport booth with an enclosed gate area. It sounds more complicated than it is: if the check is US specific, show US paperwork. If it's generic EU, show EU paperwork. Done!
Ps. I am amazed you were able to gain citizenship while you were illegal but the past is past and it doesn't matter any more.
In Europe, show your European passport. In the US, show your US passport - always... when going in or out of the country/region, show only (ONLY) the "best" passport for the situation. Basically, you won't have any more stamps or visas to worry about.
Think if it this way: it's called a passport for a reason, it lets you pass; if you have European, you are always allowed (pass) into and out of Europe and if you are a US citizen, you are always allowed (pass) in and out of the US.
I say only show one passport because you don't need to answer 1000 questions as to how you came to have two citizenships every time you go through border control. It is not illegal (actually, I think it may be in some cases, like some countries don't support dual citizenship), but border control are always on the lookout for fake passports and anybody who has more than one becomes suspicious...
Source: I too have double citizenship, I show my European passport when going in or out of Europe (and US for that matter) and Brazilian passport when going in or out of Brazil.
EDIT: When the airline asks you for your passport, they ask on behalf of the country you are traveling to, so you show/give the best one for the destination.
EDIT 2: In other words: nothing will happen to you; you no longer need visas in Europe. For a super mega complete answer; check out this question (pointed out by Henning Makholm): I have two passports/nationalities. How do I use them when I travel?