Midfielder Goodwin left Stockport at the end of the season, after the Hatters admitted they could not afford to keep his services.
And Goodwin sought advice from fellow contryman Byrne before committing himself to the club.
He said: "I spoke to Cliff about the move, he told me it is a great club and could not speak highly enough about the place.
"Looking round it is a great club, it wasn't adifficult decision to make as I had my heart set on coming here.
"I had other opportunities, but I was advised that to work with Brian Laws was the right move. I knew it was the right move for me."