Yes, only saved Christians can choose to do good in God's eyes. Lost people can choose to do good in our eyes, but not God's.
Let's say I am a lost person, and on Monday I decide to help a little old lady across the street. Then on Tuesday, I meet Harley, and he lovingly preaches the Gospel of Christ to me. I am immediately convicted and fall to my knees, a broken man. I earnestly pray to God to forgive my sins and ask Jesus Christ to come into my life as Lord and Savior. I then turn to Harley and say: "Thank you Harley for leading me to Christ, without what you did for me I might have died a lost man". Harley then slaps me upside the head and asks if I have been listening to anything he has told me. :) Anyway, then on Wednesday, I decide to help the same old lady across the street again.
Now, I would say that on Monday I did nothing pleasing to God, but on Wednesday I did. A major difference is that on Monday, I did not help in order to glorify God, but on Wednesday I did. We as outsiders would not notice the difference, but to God there would be a huge difference. Slaves to sin (the lost) cannot do anything because Christ is not in them. I think this is the sense of:
John 15:5 : "I am the vine; you are the branches. If a man remains in me and I in him, he will bear much fruit; apart from me you can do nothing.
Now, I would say that on Monday I did nothing pleasing to God, but on Wednesday I did. A major difference is that on Monday, I did not help in order to glorify God, but on Wednesday I did
If, in both cases, you helped her out of selfless compassion, because you loved her as your self, did you only choose to do good on Wednesday?
LOLOL.
"In your patience possess ye your souls." -- Luke 21:19
Works is a very complex issue. However, I believe there is only one work of God, that you believe in Christ who God sent.
The third criteria for works is that it glorifies God:
All Christians bear fruit to varying degrees. What isn't obvious is to what extend Christ uses people to "bear much fruit" or what that means. If I help an old lady across the street, I sincerely doubt if people will praise God over it. However, that old lady might be grateful in such a way as God is glorified. It's difficult to tell since God is glorified in many ways.
A non-Christian can be used of God to bring glory to Him such as Judas or Balaam, satisfying point #2 and #3. However, they do not please God simply because they do not believe upon the Son, point #1, and so fail the criteria.