I believe that God expects us to physically take care of our bodies.
I also believe God expects us to seek Him, first and foremost.
When I first met God I didn't understand a lot about theology, prayer, or the Great Commission. I just knew I was a sinner in need of forgiveness. As I grew and learned, I came to struggle with a lot of things in my life that were causing me (and my friends) to stumble. I gradually changed a lot of my habits, and learned how to walk closer to how Jesus walked- not that I'm an expert, mind you.
I didn't get interested in my health until about 5 or 6 years ago. Before that, I was overweight, out of shape, and a typical Cajun guy. I ate everything fried, double helpings, with tons of sugar and junk on the side. High cholesterol, high blood pressure, borderline diabetic. I felt like crap and my mental health began to suffer because of my self-inflicted physical hardships.
I couldn't say it was 100% God's direction that made me decide to get healthy, as much as it was making up my mind to prove I could do it to myself and my friends and family. Whatever the motivation though, it worked, and I started then what was to be the turning point in my health. I'm in better shape now in my 40's than I was in my 20's and 30's. The same wedding ring that fit tightly when I was 22 would fall off if I shook my hand. I finally got my wedding ring resized down a full size, last month, on my 19th wedding anniversary. It's still kind of loose, actually.
But in the last 5 years I've also fought with the personal struggle to keep my desire for fitness from becoming a spiritual idol in my life. When I was riding my bike 100 miles a week, and gathering acclaim and praise for it, fitness became the driving force in my life. It dictated how and when I ate, when I slept (if at all), how I spent my money, and I meditated on it day and night. It overshadowed everything else, including my family and my spiritual walk, I'm ashamed to say.
Then I got hit by a car in July of 2011, and I spent 3 months in rehab after having my femur reconstructed. And as I spent weeks in bed, unable to move, utterly helpless and strung out on painkillers, I realized that it could have been avoided if I hadn't pushed myself daily to the brink of exhaustion. After a year of blindly running myself ragged for the approval of men, God had gotten my attention in the most crippling way possible. I begged God to forgive me for ignoring Him, and effectively, my walk with God got a reboot. I read more scripture in those 3 months than I had in years previously because I literally couldn't do anything else.
I don't expect new believers to want to take care of their bodies the way that I do now if God hasn't brought them to that point yet, just like it took me years to come to grips with the self-harm I was inflicting on myself. Is it important? Yes. Is it critical to your faith? No. God can use you however He wants, regardless of your BMI. But the truth is, sin is sin, and I believe not taking care of yourself (to as much of an extent as possible) is a sin. It is no different than neglecting your prayer life, or neglecting your tithing, or neglecting to hedge your mind against temptation.
Being fat isn't a sin, but refusing to care for yourself as best you can (for whatever selfish reason) is a sin. As always, it is the motivation, the heart, that God is interested in. "It's not what goes into a man's mouth that makes him unclean, it is what comes out that makes him unclean" (Matthew 15:11). If you use self-neglect to hide from God as a coping mechanism, and live in a self-inflicted temple of pity, then God can and will break through that to get your attention, just as He will break your desire for human approval in the same way.
If you are feeling guilt and conviction about it, then maybe God is trying to tell you something. You should honor the temple (your body, see 1 Corinthians 6:19-20) that God has given you, and be a good steward of what you have. But be warned: it is a lifelong process, much like any other conviction. Be prepared to struggle, have setbacks, and eventually, you will be closer to God. And just like any other struggle with sin, don't expect it to be easy.
Just don't run to the other extreme, fall into the trap of self-worship, and end up in the same place I did. You may not even realize you're doing it- I certainly didn't. But if you feel God is telling you to deal with it, then you'd better listen.
Don't do it because people make you feel guilty. Take care of yourself because it's the right thing to do.