I read the responses with interest because I am going through the same thing with my 4 year old son AND I am a Biology teacher!
I gave him a little bug catcher/magnifier. We have looked for rolly pollies, lightning bugs, and ants. He is getting better with the crawling bugs. I spend time explaining to him how the bugs are helpful (pollinating flowers, spiders eat many flying bugs, etc) and sometimes I make up some of their names and jobs. :)
One time he saw a flying bug and asked what it was...I had no idea where it was or what it was doing so I called it a rainbow sugar snap. That just changed his whole perspective and we could talk about how 99% of the bugs are not going to even bother him, and that they are just as scared of him. Now, I don't advocate lying to children, but just making up the sweet name completely changed his thought and made him more reasonable.
The screams have not completely stopped, but they are calming faster and are less frequent. As difficult as it is, dealing with their fears means facing their fears. That is a great life lesson for the future anyway. I am going to take the bug 'o week idea for the rest of the summer. (It might be 2-3 bugs per week!) Good luck.