Immediately we recognize a delicate balance between our love for our families and our love for our Savior. Putting it in a very simple form: Jesus is to be First in our love; our families come after Him! No one is to receive more love than our God. That's why Jesus said the greatest of all commandments is to "love the Lord your God will all your heart, soul, and mind. Now, doing so is very practical. When you love the Lord as you should then, and only then, will you be able to love others the way you should. If God is not first place in your life, then no one will be in their proper place in your life!
In our Daily Verse today Jesus is putting verses 36 and 37 together. What He is saying is, "don't let your love for your family keep you from coming to love Jesus." And that's where the issue of your family becoming your enemy comes into play. Some people actually face the situation of a family member or members who warn the individual that if they believe in Jesus as their Savior, the family will "disown" them or even worse. The individual will have to make the choice, staying in harmony with their family or risking their hatred but turning to Christ and receiving Him as their Lord and Savior.
It's not that Jesus expects you to not love your family. Actually what He wants you to know is that when you turn to Him and love Him with all your heart, you will then be able to love your family as you should! Your family may not recognize that at first, but if you love them with a 1 Corinthians 13 kind of love, they won't be able to miss that you haven't turned against them.
Please note John 7:5 NKJV - "Even His brothers did not believe in Him." That's referring to Jesus' brothers before His crucifixion and resurrection. It doesn't say they hated Him, but they weren't agreeing with Him. So Jesus Himself knows what its like to have some family members being in opposition to Him. (Eventually we know that at least two of His brothers, James and Jude, did become believers. They're the ones who wrote the New Testament books: James and Jude!)