I have a lot of experience in this field So I am sure about what I told you. You can travel back into the US with an expired green card IF you can show that you have not been outside more than a year. :)
For confirmation, you can read about it here: http://barbados.usembassy.gov/legal-perm-residents.html
Here is the excerpt from the US embassy website (it's for Barbados but they are all the same):
My Green Card expired. I have been out of the United States for less than 12 months. Do I need a transportation letter?
If your permanent resident/Green Card expired and you are outside the United States, you can still return without a transportation letter if you have not been outside the United States for more than 12 months.
1)Can you naturalize with an expired green card?
Once you enter the US, you can submit your application to naturalize if you are eligible. USCIS has issued different guidance throughout the years on whether you can do this with an expired green card. In short, having an expired green card should not affect your citizenship
application. The main issue is that you will not have a proof of your status while your citizenship is pending if you dont renew your green card. It could be difficult to travel or prove your status for employment etc. That is why USCIS does recommend that you do renew your card if it is expired when you apply for citizenship. It is safer to renew it first.
2)I was out for 11 months in 2013 into 2014, I read you can only be 6 months or less unless certain circumstances apply.
Being out of the US more than 6 months in the last 3 years does not bar you from citizenship completely but you will have the burden of proving that you never abandoned your residence here in the US. To do this you would have to show you still things such as a home here, car, bank account, filing tax returns. You don't need all, but the more you show the better the chances will be. Any trips of more than 12 months will bar you from applying and your 3 years starts starts again the day you arrived back in the US from the 12 month trip.
3)Can I file from overseas? No you can only do this under certain circumstances and it doesn't sound as if you qualify.
It must be your US citizen spouse that is employed abroad under one of these jobs.
Qualifying employment abroad means to be under employment contract or orders and to assume the duties of employment in any of following entities or positions:
•Government of the United States (including the U.S. armed forces);
•American institution of research recognized as such by the Attorney General;
•American firm or corporation engaged in whole or in part in the development of foreign trade and commerce of the United States, or a subsidiary thereof;
•Public international organization in which the United States participates by treaty or statute;
•Authorized to perform the ministerial or priestly functions of a religious denomination having a bona fide organization within the United States; or
•Engaged solely as a missionary by a religious denomination or by an interdenominational mission organization having a bona fide organization within the United States.
In conclusion, unless you were inside the US at least 1.5 years TOTAL in the last 3 years with no trips of more than 12 months and proof for the trip of 11 months (like I mentioned above) then I would suggest abandoning the LPR and reapplying later. If you think you qualify for citizenship you can apply but remember you will have to be returning to the US for the fingerprints and then the interview.
Now that I have answered your follow up questions, I kindly ask that you remember to issue a rating.
Any bonus that you wish to leave is also very appreciated. Thank you!