Warning! Spoilers for Justice League Annual #1.
Although its appearance is brief, Hawkgirl’s new costume means that she doesn't just act like Wolverine, she looks like him too. She receives it after being spirited away to the future in Justice League Annual #1. The issue was written by Brian Michael Bendis and illustrated by Sanford Greene.
In the future, the militarized hero O.M.A.C. discovers Hawkgirl frozen in time, encased in a weird bubble. He is then pulled through time himself to the present. After a brief interaction with the Justice League, O.M.A.C. recognizes Hawkgirl and wrongly believes she is responsible for his time traveling. In reality, the time anomaly has been caused by a Timelord. While battling the Justice League, the Timelord tosses a weapon that freezes Hawkgirl in time and then sends it to the future. This is the strange encasement O.M.A.C. discovers her inside at the beginning of the issue.
It's not shown in the issue, but O.M.A.C. discovers her in the future, frees her, and together they work to take down an evil future corporation. This is all explained by Hawkgirl when she miraculously returns to the present just a few seconds after being sent away. When she comes back she is wearing a new set of armor, presumably something she gained during her adventures with O.M.A.C. in the future. This armor is more stylish and warrior-like, with a strange resemblance to Marvel's Wolverine.
It's fitting that Hawkgirl's new look puts her in line with Wolverine as she already shares many similarities with the character. She is known for her primal, almost feral-like rage when she attacks, just like Wolverine. She fights with brutality using her claws, which Wolverine is also famous for. The similarities continue with their relationship to unique metal. For Wolverine it's adamantium. For Hawkgirl it's Nth metal.
What they have in common also extends to their behavior in relation to their respective teams. Although a part of the X-Men, Wolverine is known for being a lone wolf and often going on his own solo missions. A great example of this is during Marvel’s Civil War. While all of the heroes are busy fighting each other, Wolverine decides to go after the villain Nitro instead. And it just so happens Hawkgirl is sent on her own adventure in order to acquire this new armor. She experiences years’ worth of fighting in the future in just the span of a few seconds in the present. If Hawkgirl isn't the Justice League's Wolverine, then nobody else comes close.