If I lost both my naturalization certificate and ...

I wish I knew the answers.  If you are here legally, a good place to start in at the Department of Motor Vehicles. 

If you have a Valid Driver's license and current utility bills in your nam, that's a good place to start.   

If you are married, bring you marriage license, if you have kids, bring their birth certificates.  

Anything that will prove you are in the US legally. 

If you are illegally in the United States; that's another story! I do not know but I would check the internet of immigration and Naturalization.  

If you are Hispanic, I bet they will give you a hard time, so I would try to find an online answer.

I knew a White Guy from Denmark who stayed in the U.S. on a Student Visa for about 20 years (and that's been a long time ago).  

Trust me, Lady liberty doesn't wear a blindfold anymore.  

Good luck!

QT 

There is. Two options:

First, if you previously had a passport but lost/ had it stolen, then you should apply for a new passport, along with the lost/ stolen form (DS 68) AND request the passport office to conduct a "File Search" when processing your application. There is an additional fee of $60 for this and you have to attach a statement to your application explaining why you have no other easier means to prove citizenship. Explain that you lost your cert. of naturalization, that it takes an excessive amount of time to replace and that you need proof of citizenship for travelling, DMV, work etc. Nothing too long, just a concise statement as to why you cannot prove citizenship through a conventional method. Don't forget to take a long with you a valid Photo ID.

Second option is to file N-565 (replacement certificate form) and then make an InfoPass appointment and request from USCIS a "verification letter" of your citizenship. I've never done this, but from others' experience you will almost certainly have to fight them for this letter. This is the most burdensome option and not recommended unless you have never applied for a passport.