Should I put some glue in the pre drilled hole before installing an eye screw?
Are you using an eye bolt or an eye screw?
 OR         

For an eye screw, glue won't do much and is not needed. (Although some glues may act as a lubricant when you first screw-in the eye-screw.)  Use bar soap or beeswax as a lubricant.

For an eye bolt, especially if you want it to be able to pivot, use appropriate washers, plus two nuts with a lock-washer in between.  For extra reliability use Loctite, not glue, where the nuts and eye-bolt meet. 

If you don't need/want pivoting action, then one nut may suffice, but Loctite is even more important because such eye-bolts always loosen their compressive grip on the wood over time.