Did Captain Needa die in The Empire Strikes Back?

He's dead, Jim.

The script confirms his death.

Clutching desperately at his throat, Captain Needa slumps down, then falls over on his back, at the feet of Darth Vader. Two stormtroopers pick up the lifeless body and carry it quickly away as Admiral Piett and two of his captains hurry up to the Dark Lord.

Additionally, the 25th anniversary novelization confirms it.

A pair of white-armored stormtroopers lifted Captain Needa’s lifeless form from the floor of Darth Vader’s Imperial Star Destroyer. Needa had known that death was the likely consequence of his failure to capture the Millennium Falcon. He had known, too, that he had to report the situation to Vader and make his formal apology. But there was no mercy for failure among the Imperial military. And Vader, in disgust, had signaled for the captain’s death.

_{Lucas, George; Kahn, James (2011-06-28). Star Wars: Trilogy (25th Anniversary Collector's Edition) (p. 402). Random House Publishing Group. Kindle Edition.}

As an explanation of his legs moving after death, the real explanation is that the actor probably just wasn't very good at playing dead. In-universe, his body may have just been spasming a bit.

Unlike some of the other characters who were killed during the OT, but who actually survived in the EU (looking at you, Boba Fett), the EU never referred to Needa having actually survived, or being somehow revived after his death in Empire.

He ded

The (Disney Canon) factbook Ultimate Star Wars confirms his demise.

Lorth Needa is a ruthless officer who serves the Republic in the Clone Wars when Chancellor Palpatine is kidnapped by General Grievous during tne Battle of Coruscant. As commanding officer of the Imperial Star Destroyer Avenger, he takes part in the search for the rebels' secret base. After losing the Millennium Falcon while pursuing it through an asteroid field, Darth Vader Force-chokes him to death.

As does the new (again, Disney Canon) junior novelisation

The grip on his throat is no less painful for being invisible. The air stops flowing to his lungs. Blood ceases circulating in his brain. His eyesight fades to white….

"Apology accepted,” Vader murmurs, and turns back to the bounty hunters. Stormtroopers swoop in to dispose of Captain Needa’s body.

_{^{The Empire Strikes Back: So You Want to Be a Jedi?}}

And the original junior novelisation

As Vader choked him from afar, he clutched desperately at his throat. But it was no use. He dropped to his knees, lifted his head, and tried to rise, then collapsed at the feet of Darth Vader.

Vader said, “Apology accepted, Captain Needa.”

As two black-uniformed Imperial soldiers lifted Needa’s lifeless body and carried it from the bridge, the Sith Lord walked to the nearby command console, where Admiral Piett and his aides were examining data. Seeing Vader approach, Piett stepped away from his console and stood at attention.

_{^{Star Wars Episode V: The Empire Strikes Back (junior novelization)}}

He died

To the best of my knowledge, there are no further appearances by Captain Needa in the EU (now Legends) postdating his apparent death. The Wookieepedia article on Needa, generally a fairly comprehensive reference for a character’s appearances, seem to concur, insofar as it does not mention anything (Canon or Legends) that would suggest that Needa survives. He does, however, have a cousin, who appears in works set after his death.

It’s probably worth noting that the canon position seems to be that Needa is very much dead:

The grip on his throat is no less painful for being invisible. The air stops flowing to his lungs. Blood ceases circulating in his brain. His eyesight fades to white….

“Apology accepted,” Vader murmurs, and turns back to the bounty hunters. Stormtroopers swoop in to dispose of Captain Needa’s body.

_{So You Want to Be a Jedi?}