According to the Gospel of John, on which day of the week was Jesus crucified? John 19:14-16:


  "Now it was the day of preparation for the Passover; it was about the
  sixth hour. And he said to the Jews, “Behold, your King!” So they
  cried out, “Away with Him, away with Him, crucify Him!” Pilate *said
  to them, “Shall I crucify your King?” The chief priests answered, “We
  have no king but Caesar.”
  
  So he then handed Him over to them to be crucified."


Which day of the week was the 'day of preparation for the Passover'?

A good answer will give a day of the week, or explain why it is not possible to give one.
There are some difficulties with the chronology of Christ's trial and crucifixion, so I definitely cannot give you a specific day of the week and say it's certain. John 19:14 (EMTV) says, "Now it was the Preparation [Day] of the Passover ...". The other gospels do not state what day it was before the crucifixion, but all four gospels do mention it later, when Jesus' body is removed from the cross.

John 19:31 says  "it was the preparation [day]" and "that Sabbath was a high day", presumably meaning it was the first day of the Feast of Unleavened Bread. Mark 15:42 and Luke 23:54 both mention the preparation day. Matt 27:62 uses the rather odd expression, "the next day, which is after the Preparation [Day]" to refer to the feast day itself.

It is clear that Jesus was crucified the day before the feast of Unleavened bread. Traditionally it has been believed that it was Friday, since Saturday is always a Sabbath. The gospels are consistent with this, assuming John was pointing out that the next day was not only a Sabbath, but also the day of the Feast, which is kept as a Sabbath regardless of the day it falls on.

The gospels are also consistent with the Feast falling on a different day of the week, since it would still be appropiate to call the day before a "preparation day". Some Christians have proposed either Wednesday or Thursday as the day of crucifixion, but the gospels do not explicitly tell us what day it was.

Apparently the Jews in the first century thought of the three spring feasts as parts of a single celebration, which it sort of is (Passover, killing of the lamb in the afternoon; Unleavened bread, begun with eating the Passover lamb right after sundown; and Firstfruits on the Sunday within the week.

Argument for Friday only

Since I originally wrote this, I found a book by Samuele Bacchiocchi, "The Time of the Crucifixion and the Resurrection". This online chapter claims the Greek word for "preparation" always meant Friday, and would not have been used for a Special Sabbath on another day. If Greek language studies support his viewpoint, then the traditional Friday date becomes more probable. I am not qualified to evalute this.

My original conclusion

The traditional idea of a Friday crucifixion assumes that the Sabbath they were preparing for was the weekly Sabbath, and presumably also the first day of the Unleavened Bread feast. While that is possible, it is also true that the first day of the Unleavened Bread is always a Sabbath, regardless of which day it falls on. The passage you quote from John's gospel is therefore consistent with the named Sabbath being any day of the week.

Presumably Christ was crucified on Nissan 14, the day originally specified as God's Passover day, the day the Passover lamb was to be killed. The Feast of Unleavened bread began right after sundown, which would be the beginning of Nissan 15.

Theoretically, Nissan 14 could have fallen on any day of the week. There is a historical tradition that it was Friday, and that should not be ignored. If Nissan 14 fell on Thursday, then the 15th would be the special Sabbath and the 16th would have been the regular weekly Sabbath. In this case, we would expect the women to wait two days before returning to the tomb. If the 14th were earlier in the week, it is not clear why the women wouldn't have returned to the tomb on the 16th.