Hmmm, well, Paul said he was a Jew and did so proudly in Acts 22. And we know he went and visited pagan temples in Athens. Obviously, he didn't worship any but God alone, but he certainly didn't deny his Judaism or avoid steer clear of pagan temples.
Paul's trip to the pagan temples was for the intent of preaching and spreading the gospel. It does not apply to webdog's question.
The example given about Christian Muslem's going to the mosque to hide the fact that they are Christian is considerably different than Paul.
I really believe it has to be considered case base case. Situation by situation.
Example: Do I stop and share the plan of salvation with everyone I see? Of course not. But I am required to do so when I feel that God want's me to.
Too often we try to make a flowchart of Step 1 , Step 2, etc. The Gospel isn't like that. You and I may be in the same city, under the same circumstances. God may want you to stand and may direct me to flee. Or visa-versa. Elijah prayed down fire from heaven, and defeated an army, then fled from Jezabel.