"Should Christians stay away from people who aren't Christians?"

In John 17 Jesus says a prayer for all Christians.  This is what He says.  "My prayer is not that you take them out of the world but that you protect them from the evil one. They are not of the world, even as I am not of it."  (John 17:16-17)

What Jesus is saying is that we shouldn't keep away from people -- we couldn't even if we wanted too.  That said, Jesus explains that we are not of the world.  In other words we do what the Bible says we should do.  This is much different than what the world says we should do. 

So if people are doing things that keep us from our relationship with Jesus we should stay away from them.  If they don't keep us from Christ then we should be friends with them.  To both groups we should pray that people would see Jesus when they see us.  I have a study on how to do that.  I'll send it to you.

Here is some more of what Jesus said on the subject...

"You are the light of the world. A city on a hill cannot be hidden. Neither do people light a lamp and put it under a bowl. Instead they put it on its stand, and it gives light to everyone in the house. In the same way, let your light shine before men, that they may see your good deeds and praise your Father in heaven." (Matthew 5:14-16)

In those verses Jesus is comparing a Christian to a light.  We have Christ in our lives and He is saying that we should "let our light shine" or another way of saying it is when we are around people they should see the light of Jesus in us and be drawn to Him.  Or still another way of looking at it is this:  Everyday you represent the Living God to a dying world.  Let your friends see Him in you.

Now we have to address 2 Corinthians 6:14 which says, "Do not be yoked together with unbelievers. For what do righteousness and wickedness have in common? Or what fellowship can light have with darkness?"

Basically that verse is what I said above.  If someone is keeping you from Christ stay away from them because you relationship with people is temporary but your relationship with Christ lasts forever.