While /u/RingGiver is correct that Roman citizens were not normally crucified, there were incidents where this didn't hold up.

More directly, Jesus simply had no claim to citizenship. He was born under Herod the Great's rule of Judea as a client kingdom, and there is simply no grounds on which he would hold Roman citizenship.

Almost certainly not.

According to the New Testament, Jesus was scourged and crucified in circa 30-33 CE. IAs early as 509 BCE and as late as 299 BCE Rome had enacted the Valerian Laws which prevented a Roman citizen from being sentenced to death by either scourging or crucifixion without being able to appeal to the people in a formal tribune known as provocatio. (History of Rome..., Duruy Colin 2. 341-342)

These laws were expanded between 199-184 BCE in a series of laws known as the Porcian Laws which prevented the scourging of Roman citizens altogether and allowed a Roman citizen to escape a death sentence by voluntary exile. (A commentary on Livy, Oakley, x.9.3-6)

Jesus, according to Josephus and the New Testament (Mark 15, Josephus XVIII.3.3), was presented to the Roman governor and did not have (a.) the opportunity to provocatio and (b.) was in fact both scourged and crucified.

These two factors, from these particular sources, make a fairly conclusive case that Jesus was not a Roman citizen. (Wikipedia has great simple articles on Wikipedia Procian Laws and Valerian Law)