Persecution is part of the cost of living a Christian life. Jesus was communicating to this man that not having a place to stay at times was part of that cost.
There are many forms of persecution. Having your life threatened because of your faith in Jesus is one way you can be persecuted, but it is not the most damaging. History shows that the Church has always flourished under persecution with increased numbers and zeal. During intense, life-threatening persecution, people's priorities get straightened out and the Lord assumes His rightful place. This always works for our good, regardless of what our outward circumstances might be.
A far more deadly form of persecution is men simply speaking evil of you or separating you from their company. It's more deadly because it's more subtle.
Many who would never directly deny the Lord will fall into self-pity or strife because of someone's criticism. This will render one just as ineffective as a negative reaction to having one's life threatened would.
It helps to recognize that it is not you that they are persecuting, but rather Christ in you. You are actually becoming a partaker of His sufferings and will share His rewards. With this in mind, we can actually shout and leap for joy in persecution!