According to the Bible, yes, only true Christians are capable of pure, unconditional love. However, this is a tricky statement, because if you read the Bible and follow it closely, it suggests that this level of Christianity is unattainable by humans, because we have sin in our hearts. We would need to overcome our Earthly desires and love God above all else in order to be capable of this level of love. So basically, only true Christians would be capable of pure love, and there is no human alive that can call themselves a true Christian.
I don't believe that Christians are the only believers who are capable of feeling Love.. God loves as all, no matter who we are. And in the same way everybody is capable of love. As God has demonstrate He's love to us and the some form we demonstrate our love to the people who surround us.
This fact implies that not only do we have a soul and mind like God but we also possess certain characteristics that God has given us. These characteristics are what distinguishes us humans from animals. I believe that one of those characteristics is the capacity to love not only ourselves but also others and God. At the very least, I'm willing to bet that most of us love ourselves.
Since this topic is based on supposed biblical evidence, I must refute such a claim based on scripture. Matthew 7:9 says
"What man among you, if his son asks him for bread, will give him a stone? Or if he asks for a fish, will give him a snake? If you then, who are evil, know how to give good gifts to your children, how much more will your Father in heaven give good things to those who ask Him! "
Not everyone is a Christian, and just because you're not doesn't make you a bad person or incapable of love. And if you are a Christian, you're not automatically a higher form of a human being than the rest of us. We've all met good and bad Christians, good and bad atheists, good and bad Muslims, etc.
"Anyone who does not love does not know God, because God is love" 1 John 4:8
I think that you don't have to call yourself a Christian to know love. If God is love, to know love is to know God. You can argue what the bible means regarding this but I know people who would never call themselves Christians who know love more than most Christians I have met. I say if you're willing to die for someone other than yourself, that is true love and that is God's love shown to us by Jesus.
The question that remains: "What is a Christian?"