It is sad that your Christian friends do not like it when you do not gossip.  Gossip is literally talking negatively about someone behind their backs.  Whether it be true or false, talking about someone in a way that will hurt them is wrong and is condemned in scripture.  If someone has done or said something that they should not have, our duty is to speak to them face to face about their actions.  I have often said that if I have done or said something inappropriately, speak TO me, not ABOUT me. 

James 4:11 tells us, “Speak not evil one of another, brethren. He that speaketh evil of his brother, and judgeth his brother, speaketh evil of the law, and judgeth the law: but if thou judge the law, thou art not a doer of the law, but a judge.”  The Apostle Paul spoke of how he was “being defamed” (1 Corinthians 14:3), which shows that others were speaking bad, or gossiping about him in order to ruin his testimony for Christ.  I am so thankful that you do not gossip, even though you are criticized for it.  I can imagine your friends who love to gossip, also gossip about you when you are not around.  I encourage you to continue in your refusal to speak evil of others.  Deuteronomy 31:6 says to you, “Be strong and of a good courage, fear not, nor be afraid of them: for the LORD thy God, he it is that doth go with thee; he will not fail thee, nor forsake thee.”

Sometimes, we can only have a closer relationship with others if we compromise our Christian stand.  “Be strong!”  Do not allow that to happen to you.  Romans 12:18 tells us, “If it be possible, as much as lieth in you, live peaceably with all men.”  When dealing with others, especially those who are not saved, it may not be possible to have a good close relationship.  We should show the love of God to all, but we cannot be close friends with everyone.  Proverbs 18:24 warns us, “A person of too many friends comes to ruin…” (NASB).  If our intent is to have a lot of friends, we might find ourselves compromising our standards and beliefs to get and maintain those friends.  You asked how you could “build closer relationships with them and non-believers while honoring God.”  While the Lord does want you to share His love to all, He also wants you to maintain a pure testimony of Him.  As for having close relationships with unbelievers, the Lord tells us in 2 Corinthians 6:17-18, “Wherefore come out from among them, and be ye separate, saith the Lord, and touch not the unclean thing; and I will receive you, And will be a Father unto you, and ye shall be my sons and daughters, saith the Lord Almighty.”  We often have to separate from bad influences so that our relationship with and testimony for the Lord is not compromised. 

To have a close relationship with unbelievers, or Christians who are not following the Lord, you will have to establish some common ground other than the Lord, Himself.  This is not what the Lord wants.  He desires to have a close relationship with all that are saved.  In your relationship with others, be careful to keep your testimony of the Lord without compromise.  Remember…you will influence all those you are close to, or they will influence you.  (474.4)