Teams have the option to extend a one year offer, called a tender to their restricted free agents. Failing to do so as the Jets did to Mulligan, Cole, and Westerman means they can go where they want. In other words, the Jets do not want them. That is no surprise. None of those guys made a big positive impact, and Mulligan made an extremely negative one.
By giving Maybin a tender, the Jets put themselves in a position to retain him. Maybin is a free agent, but the Jets have the right match any contract to keep him. If Maybin signs elsewhere, and the Jets do not match the offer, they lose Maybin but get the other team's first round pick. Nobody is giving up a first rounder for a situational player like Maybin so he is going to sign that one year offer to stay a Jet.