Self-denial is not exclusive to Christianity. When asked to sum up the meaning of life in three words or less, Gandhi responded cheerfully, 'That's easy: Renounce and enjoy'. One newspaper report said this about the Mahatma ”The key to his daring achievements lies in his own ongoing [self-inficted] suffering, including his poverty, celibacy, arrests, imprisonments, attacks, and assassination....”.
The kind of self-denial that Jesus is calling His disciples to in this passage is different. It is a self-denial that unbelievers can’t actually practice. Why? Because it’s a self-denial that is based upon the Gospel. It's only enabled by grace and it has as its goal something only a believer can strive for or hope for.