I am a few months behind in mortgage payments. Can I file Chapter 7, not Chapter 13, and keep my home?

Only if you could pay all the mortgage arrears (back payments) at one time, which is very unlikely or else you wouldn't be in your current situation. If that's not possible, the mortgage holder could successfully ask the judge to remove the "stay" (protection) that prevents them from foreclosing on your home.

In a Chapter 13, you get to include the arrears in the monthly payment plan and pay it off over three or five years.